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X 
 
AN 
 ELEMENTAET TEEATISE 
 
 ON THE 
 
 ? • ■ ' \r^ 
 
 THEORY OF DETERMINANTS. 
 
 A TEXT-BOOK FOR COLLEGES. 
 
 PAUL H. HANUS, 
 
 M 
 Formerly Professor of Mathematigs in the University of Colorado; 
 NOW Principal of Denver High School, District 
 No. 2 (West Denver). 
 
 UHV 
 
 BOSTON: 
 PUBLISHED BY GINN AND COMPANY. 
 
 1886. 
 
Entered, according to the Act of Congress, in the year M86, by 
 
 PAUL H: IIAXUS, 
 
 in the Office of the Librarian of Congress, at Washington. 
 
 J. 8. Gushing & Co., Pkintkrs, Boston. 
 
PEEFAOE. 
 
 The importance of a knowledge of Determinants to all 
 who extend their reading beyond the elements of mathematics, 
 and the fact that most modern writers employ the determinant 
 notation, have led to the belief that an American work on 
 Determinants might satisfy a growing demand. 
 
 This is a text-book, and not an exhaustive treatise. Enough 
 is given, however, to enable the student to use the determinant 
 notation with ease, and to enable him to pursue his further 
 reading in the modern higher mathematics with pleasure and 
 profit. 
 
 The book is written with reference to the wants of the 
 private student as well as to the needs of the class-room. The 
 subject is at first presented with great simplicity. As the stu- 
 dent advances, less attention is given to details. More than 
 half the volume is devoted to applications and special forms, 
 that the reader may get some notion of the power and utility 
 of determinants as instruments of research. 
 
 Throughout the work care has been taken to show how each 
 new concept has been evolved naturally ; and, whenever it is 
 thought advisable, a special case precedes the general dis- 
 cussion. 
 
 The work has been written in the far West, where contact 
 with others in the same field was practically impossible. I 
 
 167219 
 
iv PREFACE. 
 
 shall therefore be grateful for any notification of errors that 
 may have escaped detection. 
 
 My thanks are due to Messrs. J. S. Gushing & Co., of 
 Boston, for great care and patience manifested in the prepara- 
 tion of the plates. 
 
 Among the works consulted most assistance has been derived 
 from the following. All the works named have been used 
 freely. 
 
 Matzka. — Grundziige der systematischen Einfiihrung und Begriin- 
 dung der Lehre der Determinanten. 
 
 Baltzer. — Theorie und Anwendung der Determinanten (Fiinfte 
 Aufiage). 
 
 Gunther. — Lehrbuch der Deterniinanten-Theorie (Zweite Aufiage). 
 
 Diekmann. — Einleitung in die Lehre von den Determinanten und 
 ihrer Anwendung auf, etc. 
 
 Dostor. — Elements de la Theorie des Determinants avec Applica- 
 tions, etc. (Deuxieme edition). 
 
 Houel. — Cours de Calcul Infinitesimal. 
 
 Scott. — A Treatise on the Theory of Determinants and their Appli- 
 cations, etc. 
 
 Burnside and Panton. — The Theory of Equations, with an Intro- 
 duction, etc. 
 
 Muir. — A Treatise on the Theory of Determinants. 
 
 I am especially indebted to the last, two works for many 
 examples. 
 
 PAUL H. HAN US. 
 
 BouLDBU, Col., May, 1886. 
 
co]:^te:n"ts. 
 
 CHAPTER I. 
 Preliminary Xotioxs and Definitions. 
 
 AET. PAGE. 
 
 1. Discovery of Determinants 1 
 
 2-7. Determinants produced by eliminating the unknowns 
 
 from a system of simultaneous equations . . 2-8 
 
 8-10. Values of the unknowns in determinant form . . 8-10 
 
 9. Change of sign 9 
 
 11. iiTotation 11-12 
 
 12-14. Expansions with square notation .... 13 
 
 15. Rule for expanding a determinant of the third order . 14 
 
 Examples ' 14-16 
 
 CHAPTER n. 
 General Properties of Determinants. 
 
 16-19. Definition and notation . . . . . . 17-20 
 
 20. Corollaries 20 
 
 21. Inversions of order 20-21 
 
 22. Number of terms in a determinant . . . . 21 
 23-24. Corollaries; expansions 22-23 
 
 25. If the rows in order are made the columns in order, etc. 23 
 
 26. Number of positive and negative terms ... 24 
 
 27. Interchange of two parallel lines 24 
 

 VI " CONTENTS. 
 
 AUT. PAGE. 
 
 28. Two identical parallel lines 25 
 
 29. Cyclical permutations . r • 25-26 
 
 30. Corollary 26 
 
 Examples 27 
 
 31. Every element of a line multiplied by the same 
 
 number 27 
 
 32-33. Corollaries 28 
 
 34-35. Decomposition of determinant with polynomial ele- 
 ments 28-29 
 
 36. Converse of 34 . . . . ' . . . . 30 
 
 37. Transformation by addition of parallel lines . . 30-31 
 
 38. Minor determinants, or Minors . . ... . 31-32 
 
 39. Expansion of determinant as linear function of the 
 elements of one line . . . . . . . 32 
 
 40. Coefficient of any element in the expansion of a 
 determinant 33-34 
 
 41-44. Corollaries ; expansions 34-36 
 
 Examples 3G-40 
 
 45. Elements of a line multiplied by first minors of corre- 
 sponding elements of a parallel line . . . 10-41 
 47. Expansion in zero-axial determinants .... 41-42 
 48-49. Simplification by taking each consecutive pair of ele- 
 ments in the first row, etc 43-44 
 
 Examples . . . . ... . . . 45-46 
 
 50-54. The product of two determinants .... 40-53 " 
 
 Examples 53-56 
 
 55. Laplace's Theorem (expansion) 56-57 
 
 57. Product of a determinant by one of its minors . . 58-60 
 
 58. Rectangular Arrays or Matrices (product of) . . 61-63* 
 59-62. The Reciprocal or adjugate determinant . . . 63-68 
 
 Examples 69-71 
 
 63. Special expansions (including Cauchy's Theorem, 
 
 Example III.) . . . . . . . 71-76 
 
CONTENTS. 
 
 Yll 
 
 64. Solutions of certain determinant equations . 
 65-66. Differential of a determinant 
 
 PAGE. 
 
 76-78 
 79-81 
 
 CHAPTER III. 
 Applications and Special Forms. 
 
 67-70. Solution of a system of simple simultaneous 
 
 equations 82-86 
 
 71-72. If the equations of the system are not indepen- 
 dent 86 
 
 73. If mi = m2= '•' =mn-i = 0, and one m as wi« does 
 
 not 87 
 
 74-78. Solution of a system of linear homogeneous 
 
 equations 87-92 
 
 1 i 77. The condition A = for a system of homo- 
 geneous equations 91 
 
 79. Condition fulfilled when n equations containing 
 
 n— 1 unknowns are simultaneous . . . 92-93 
 
 80. Solution of a system of simple equations by 79 . 93-94 
 
 81. Another application of 79 94 
 
 82-83. The Matrix as the result of elimination . . 95-97 
 
 84-89. Applications of preceding processes . . . 98-110 
 
 90-100. Resultants, or Eliminants 110-126 
 
 91. Euler's ]\Iethod of elimination .... 111-112 
 
 92. Sylvester's Method . . . . . . 113-114 
 
 93. Bezout's Method (Cauchy) 114-118 
 
 94-95. Resultant in terms of the roots .... 118-121 
 
 96. Properties of Resultant . . . . ' . 121-122 
 
 97-100. Applications of Sylvester's Method . . . 122-126 
 
 101-102. Discriminant of an equation .... 126-129 
 
 103. Resultant of a system of homogeneous equations 
 
 when n — 1 are linear, and one is of the second 
 
 degree 129-130 
 
VIU 
 
 CONTENTS. 
 
 ART. PAGE. 
 
 104—105. Special solutions of simultaneous quadratics . 131-136 
 
 106. Solution of the Cubic . . .... 137-138 
 
 107-112. Symmetrical determinants, definitions, and some 
 
 special properties 138-144 
 
 113. If X be subtracted from each element of the prin- 
 cipal diagonal of a symmetrical determinant . 144-146 
 114-118. Orthosymmetric determinants . . . • . 146-150 
 119-126. Skew determinants and skew symmetrical deter- 
 minants 151-159 
 
 127-131. Pfaffians 159-164 
 
 132-134. Circi:^lants 164-169 
 
 135. Centrosymmetric determinants .... 169-171 
 
 136-150. Continuants 171-186 
 
 151-163. Alternants 187-201 
 
 164-170. TheJacobian ....... 201-209 
 
 171. The Hessian 209 
 
 172-176. The Wronskian 209-212 
 
 177-179. • Linear substitution ; Orthogonal substitution . 213-217 
 

 THEORY OF DETERMINANTS. 
 
 CHAPTER I. 
 
 ' PEELIMINARY NOTIONS AND DEFINITIONS. 
 
 1. The first notion of Determinants we owe to Leibnitz, who, 
 in his attempts to simplify the expressions arising in the elimi- 
 .nation of the unknown quantities from a set of linear equations, 
 employed symbols nearlj' identical with our present determinant 
 notation. In a letter dated April 28, 1693, Leibnitz communi- 
 cates his discovery to LTIospital ; and later, in another letter, 
 expresses the conviction that the functions will develop remark- 
 able and very important properties, — a conviction which time- 
 has abundantly verified. Leibnitz, however, never pursued the 
 subject himself, and his discovery lay dormant till the middle 
 of the eighteenth century. 
 
 In 1750 the celebrated geometer, Gabriel Cramer, rediscovered 
 determinants while working upon the analysis of curves. Dur- 
 ing the course of his investigations, Cramer had to solve sets 
 of linear equations, and naturally encountered the same func- 
 tions that had attracted the attention of Leibnitz.* To Cramer 
 is due the general rule for the solution of n simultaneous linear 
 equations (non-homogeneous), containing as many unknown 
 quantities. 
 
 This rule was inferred without proof from the form of the 
 values of the unknown quantities obtained in solving sets of 
 two and three equations. 
 
 ♦ The particular problem which led to Cramer's discovery of deter- 
 minants appears to have been : To pass a curve of the uth order through 
 
 u'^ 8y . . . 
 
 any — | given pomts. 
 
 2, 2, 
 
2 THEORY OF DETERMINANTS. 
 
 Since the time of Cramer important advances hare been 
 made. The names of many celebrated mathematicians appear 
 in the list of those who aided the evolution of a theory of deter- 
 minants. Prominent among these are Vandermonde and Gauss. 
 From Gauss the name "determinant" instead of "resultant" 
 was adopted by Cauchy. Cauchy and Jacobi are perhaps to 
 be considered as the greatest among those who first developed 
 the subject. The monograph of Jacobi, published in 1841,* 
 established the foundation of a treatise on the theory of 
 determinants ; and his own writings, as well as the works of 
 many eminent mathematicians during the past fifty years, attest 
 the wonderful power of determinants as instruments of mathe- 
 matical investigation, and the fruitfulness of the functions 
 themselves. 
 
 2. The most natural way of approaching the theory of deter- 
 minants would be along the line of development. This is 
 accordingly our purpose. Owing to peculiar difficulties attend- 
 ing this mode of procedure, we can however onl}' employ this 
 .method at the outset, and must soon adopt a presentation 
 better suited to the further unfolding of the subject, and free 
 from the peculiar difficulties alluded to. 
 
 Determinants of the second^ thirds and fourth order. 
 
 3. Consider the set of four simultaneous linear equations : — 
 
 (1) a^x -{- h-^y -\- CiZ -\- d^t — rrii 
 
 (2) a<iX + h^y + c^z -j-dzt = n^ 
 
 (3) a;iX-^b.y-\-CsZ-\-dst==ms 
 
 (4) a^x -\- b^y -^ c^z -{• d^t = m^ 
 
 Here it will be convenient to eliminate the unknown quantities 
 in a uniform manner, as follows : in each set of equations to be 
 obtained, (2) will be multiplied by the coefficient of the un- 
 known in (1) that is to be eliminated, and (1) by the corre- 
 sponding coefficient in (2) ; (3) will be multiplied by the 
 
 * De Formatione et Proprietatibus Determinantium. 
 
 I. 
 
PRELIMINARY NOTIONS AND J3EF1NITI0NS. 6 
 
 coefficient of tlie unl^nown under consideration in (2) , and (2) 
 by the corresponding coefficient in (3) ; and so on tlirough the 
 set. Having tlius made the coefficients of one of the unknowns, 
 it', say, the same in all the equations, we will then eliminate x 
 by subtracting (1) from (2), (2) from (3), etc. We shall find 
 in performing these operations that the coefficients Of the un- 
 known quantities and the absolute terra after each elimination 
 are functions of a particular form, and subject to the same law 
 of formation, — that these functions are, in fact, Determinants. 
 Eliminating x in set I. as directed, we have 
 
 (1) (di^s — <^2^l)2/+(^1^2 — «2Cl)^+(<^l<^2— %^l)^=<^l*^2 — «2^ll 
 
 (2) {a,^z—aJ)<^y-\- (r/oCs— agCs)^;^ {a2dz—a.jic^t=a.2m^—a^m2 \ II. 
 
 (3 ) ( a^^ — a^dg) y + (^304 — a^c^ z + {a-^d^ — a^d.^ t = a^m^^ — a^m^ j 
 
 4. Examining these binomial coefficients, we see that each 
 contains one positive and one negative term, and involves four 
 quantities, viz., ai, a2, 61, &2 i or ag? %? <^2? Cg, etc. It will also 
 be noticed that each term never contains more than one a 
 (coefficient of a?) , or h (coefficient of 2/) , or c (coefficient of z) , 
 etc., but that each term does contain all the subscripts that 
 occur in the binomial. Finally, the terms in which the sub- 
 scripts occur in their natural order are positive, while in the 
 negative terms there is an inversion of the natural order in the 
 subscripts, ^.e., a^^c^ is +, but a4C3 is — . Such binomials are 
 determinants of the second order.* (The order of a determinant 
 is determined b}' the number of factors in each term.) It has 
 been agreed to denote them, following Laplace, by writing the 
 letters involved in regular succession, affecting each with the 
 subscripts in order, and enclosing the whole expression within 
 parentheses, thus : {a^h^) = ai^a — ^3^1 '•> {(^2^^) = «2^3 — <^3C2, etc. 
 
 Introducing this notation, set II. becomes 
 
 ( 1 ) («! 62) y 4- (or-i C2) z + (a-i d^) t= {a^m^) ' 
 
 (2) {a^h) y + (agCg) z + (ciodg) t= {cum^) . III. 
 
 (3) {a^h,) y + (otgC^) z + {a^d^) t = (^3^714) 
 
 * The general definition of a determinant is given in 17, Chap. II. 
 
IV. 
 
 4 THEORY OF DETEllMINANTS. 
 
 5. If we now eliminate y, according to the directions given 
 in 3, we have 
 
 (1) [("1&2) («2C3)-(«2W (%C,)]2+[(aiW<«2«^3) 
 
 — (agftg) (aiC^2)]^= («1&2) (^2^3) — {^M («l'^2) 
 
 (2) t (ag^s) («3C4) - («3&4) («2C3)] 2 + [(a2&3) («3^4) 
 
 — (a364) {a2dz)~\t = {a^h) {a^m^)— {a^h^) {a^m^) 
 
 Examining the binomial coefficients of the unknowns, and 
 the absolute terms in set IV, we see at once that they are of 
 the same form; and if we can simplify any one of them and 
 discover the law of formation, we have them all. For this 
 purpose let us expand the coefficient of 2;, putting, for short- 
 ness, this coefficient equal to C. Then, by the definition in 4, 
 G = (ai&2) (ci2Cs) - (a^bs) (a^c,) 
 
 = (ai^a) (^2^3 — cisGo) — (a^bs) (ttjCa — agCi) 
 = ajj [ (ai 62) C3 + (a2 ^3) ^i] - Cg [ (ai 62) ^3 + («2 ^3) «i] • 
 The last binomial, 
 
 (aj 62) «3 + (^2 ^3) «1 = («1 ^2 — «2 ^1) % + («2 &3 — «3 ^2) «! - 
 = ^2 («1^3 — «3^) = «2 («1^3) • 
 
 .-. C = a2 [(ai62) C3 — (aibs) Gi + («2&3) cj 
 
 = ttg [«! &2 C3 — ttg ^1 C3 — «i &3 C2 + ^3 61 Co + 0^2 ?>3 Cj — Cl^ h^ cj . 
 
 Here the quantit}' within brackets consists of 2-3 = G terms, 
 ^.e., of as many terms as there are permutations of the sub- 
 scripts 1/2, 3- Three of the terms are positive and as many 
 are negative. The quantity involves the 3^ = 9 quantities a^, 
 
 ^29 ^35 "H b^-, O3, Cj, C2, C3. 
 
 No term involves more than one a, or &, or c, but does con- 
 tain all of the subscripts 1, 2, 3? each term containing a different 
 permutation of these numbers. Finally, as before, we notice 
 that those terms in which the subscripts occur in their natural 
 order, or in which there is an even number of inversions * of 
 
 * In a series of integers which are all different there is said to be an 
 inversion of order when a greater number precedes a less. Thus in 13452 
 there are three inversions, in 21354 there are two inversions, etc. 
 
PRELIMINARY NOTIONS AND DEFINITION^. 5 
 
 order, are positive, while those terms are negative in which the 
 number of inversions of order of the subscripts is odd. Such 
 a function is a determinant of the third order. A determinant 
 in which the quantities are those of C is denoted by (rti&aCa). 
 We therefore have C = a2 (%&2C3) . It must be carefully noticed 
 that the equation 
 
 (tti^gCg) = (a^bo) Cs — (chbs) c^ + {ci^h) Cj 
 
 = tti^aCs — a^hiC^ — aibsC2 + (Xs^iCa + ci2^sCi — a^b2Ci 
 
 gives the expansion of a determinant of the third order. 
 
 Employing the notation just explained, the coefficient of 
 t in (1) is evidently a^^aib^d^), and the absolute term is 
 «2(ai&2W3)« The coefficients and the absolute term of (2) will 
 obviously be a^Xaob^c^), a-^ia^b^d^^ a^ia^b^m^^ in order. 
 
 Introducing this notation into set IV, and dividing (1) and 
 (2) by tta and a^ respectively, we have 
 
 ( 1 ) (ai b. C3) ^ + (ai 62 ^^3) ^ = («i h m^) , 
 
 ;;i 
 
 ( 2 ) (ttg 63 C4) 2; + (as &3 <^4) t= (a2bs m 
 
 6.* If we now eliminate z in the same manner as heretofore, 
 we have 
 
 = («1&2C3) («2&3W^4) — («2^3C4) («1&2^%) 
 
 VI. 
 
 The preceding results naturally imply a simplification and law 
 of formation to be discovered in the coefficient of t and the 
 absolute term of VI. 
 
 To simplify the coefficient of f, which for shortness we will 
 call (7, as before, we proceed as follows : 
 
 C = (ai^gCs) ((1263^4) — (<^2&3C4) (0tl&2^3) 
 
 = (aihcs) [(a2^3) d* — («2M <^3 + («3&4) (^2] 
 
 - ((hhCi) [(Ctj 62) ^3 - (CLlps) ^2 + («2&3) <^J 
 
 = {<hh) [(ai&2C3) <^4 - ( o.ihc^) d{\ - Dd., + D,d2', 
 * 6 may be omitted on first reading, if thought best. 
 
6 THEOHY OF DETERM1NA^'TS. 
 
 in which 
 
 D = {a^h^c^ (^2^4) + (a2^3C4) (%&2)? and 
 
 Now, by 5, (aidgc's) = {aib^) Cg — {aM c^ + {a^h) Ci ; 
 and (ag 63 C4) = (ag ^3) ^4 — («2 ^4) C3 + (t^s ^4) ^2- 
 
 Substituting, 
 
 D = (a^h) [ia,\) C3 - (ai&g) Cg + (agftg) cj + (aiftg) [(«2&3) C4 
 
 — («2MC3+(«3^4)C2] 
 = (as 63) [(^2^ Ci+ (Oti^s) Cj - [(a2&4) {(^ih) 
 
 -{a^h) (a3&4)]c2- 
 
 The second binomial, {a^h^) {aih^) — ((1162) (^3^4) 
 
 = {a^h^-a^\) {aM - (^3^4 -«4 ^3) ((^ih) 
 
 = &4 [(ai?>3) 0(2 - (oti&2) «3] - <^4 [(tti^s) ^2 - (^1^2) hi 
 
 = ^4 [(«1^3 — «3&l) «2 — («1^2 — «2^) ^3] 
 
 — ^4 [(tti 63 — ttg 61) 62 — (<*! h — a^bi) 63] 
 = <Xi 64 (as 63 — cca bo) — O4 61 (tta ^s — % ^2) 
 
 = (0164) (ag&g) (K) 
 
 .*. D = {a^b^) [(ai^a) C4 — (ai&i) ^2 + (a2&4) cj. 
 
 Substituting the expansions of {a^b^c^) and of {a2biC^ in Di 
 we have 
 
 A = («'3M [(«1^2) C3 - (ai?>3) ^2 + («2W Cj + («l&3) [(a2^3) C4 
 
 — («2WC3+ («3MC2] 
 
 = {a.p.,) [(a3&4)ci+ (0163)^4] - [(ai&3) («2?>4) - («A) (ot3^4)]e3. 
 
 Here we notice that the binomial factor of the second term is 
 the same as the binomial factor in the last term of D : hence 
 equation (K) above, is {a^^b^ (<^i&4)« 
 
 .-. A= {(^ih) [(«1^3)C4- («1&4)C3+ (a3^4)^i]. 
 
 Substituting the values of D and Di just obtained, in C, 
 we have 
 
PRELIMINARY NOTIONS AND DEFINITIONS. 7 
 
 O = {a^h) [_{a^h.2C^)d^— {a^h^c,)^.^— \{(hh.?)c^~ («! 64)03 
 
 = (^^s) l{.(^ihG^)d^— (a^hc^)d.i-h {chb^Ci)(l2— (aobsC,)d{]. 
 
 From this value of C the absolute term of VI is obviously 
 
 ((^kh) l(ctib2C^)mi— (ai 62^)^3+ (^i 63 04)^13— (a2&3C4)mJ. 
 
 Now the quantit}^ within brackets in C (and in the absolute 
 term) of YI is here seen to be composed of four terms, each 
 of which contains a factor which is a determinant of the third 
 order. We shall presently show that this quantity is a deter- 
 minant of the fourth order ^ and will therefore write, in accord- 
 ance with the notation already exemplified, for determinants of 
 lower orders : 
 
 {a^h2C^)d^ — {a^b.2C^)ds + («iV4)<^2 — (ct2hcd^i = (<^hhc3d4) • • • (R). 
 
 Now, 5, («! 62 Cs) = («! h) C3 - («! 63) 6-2 + (a2 63) Ci ; 
 
 (aib2C^) = (ai 62)04 — (% 64)02 + (^2 64)^1 ; 
 
 (ciibsc^) = (ai 63)04 - (0^^64)03+ ((1364)01 ; 
 
 (a2 6304) = («2 63)04— (0^264)03+ (a3 64)02. 
 
 Expanding the determinants of the second order in the 
 second members of these equations according to 4, and sub- 
 stituting in equation (R) , there results ; 
 
 (ai6203cZ4) =ai6203d4— a26i03cZ4— a.i6302f^4H- a36i02d4+ a2b^Cid^— a^^^^id^ 
 — aib^c^d^ + a^biCids + a^^c^d^ — a^-^c^d^ — a'^^c-^d^ + ap^^^d^ 
 +(116304(^2— «36i04C^2—«i64C3^2+ a46i03d2H- a^b^pid^— a^b^c^d^ 
 — a2bzC4il^-\-a^h^^di-\-a2b4p^di— a^2f-'.^di— a^b^c^di-^ ajft-^c^di. 
 
 This expansion contains 4 -S- 2 = 24 terms, involving 4^=16 
 quantities. Each term contains only one a (coefficient of x) , 
 or 6 (coefficient of ?/) , or (coefficient of z)\, or d (coefficient 
 of t) , and contains all the subscripts ; a different permutation 
 of the subscripts belonging to each term. As before, we find 
 that the number of inversions of order of the subscripts is an 
 even number in the posi^ve terms, and is an odd number in 
 
8 THEORY OF DETERMINANTS. 
 
 the negative terms. Moreover, the number of terms is exactly 
 the number of permutations of the first four natural numbers. 
 Such a function is a determinant of the fourth order, and is 
 accordingly designated by {aih^c^d^. Introducing this nota- 
 tion, and dividing by (aa^s)? equation VI becomes 
 
 {aih2Czd^t=^ (aibzCsm^), VII. ' 
 
 It is to be noticed that equation (R) of the present article 
 gives the expansion of a determinant of the fourth order. 
 
 7. We have now shown how determinants of the second, 
 third, and fourth orders arise in the solution of simple simul- 
 taneous equations. From the reductions of 6, it is obvious 
 that to continue the present method would very soon imply 
 difficulties in the simplifications practically insurmountable when 
 we attempt to produce determinants of the higher orders. For 
 determinants of the fifth order, the process of reduction would 
 be found very tedious. Hence, to investigate the properties of 
 determinants of the nth order, we are forced to take a new 
 starting-point ; and in Chapter II. we proceed upon a plan 
 somewhat different from that hitherto adopted. 
 
 Values of the Unknown Quantities. 
 
 8. From equation VII, 6, ^ = T-^-v-^rv- Had the equations 
 
 of set I been so arranged that z should be the last unknown 
 
 m each equation, we would evidentlv have z = -y — 7—5 — r- 
 ^ ,, (a,d,c,m,) ^ ^ (d,b,c,m ,) (^^^^ds^^) 
 
 In the same way, y = ^^-^^-j ; x = "p^^^^j- 
 
 9. Among the many properties of determinants to be estab- 
 lished, we may here produce the following theorem, which is 
 among the most important of the elementary theorems in the 
 subject : 
 
 The interchange of two letters, or of two subscripts, the others 
 remaining undisturbed, changes the sign but not the magnitude 
 of a determinant. ^ 
 
VALUES or THE UNKNOWN QUANTITIES. 9 
 
 1st. For determinants of the second order. 
 
 (a) The interchange of two letters. 
 
 {aih.2) =ai62 — ^2^1- In this, if we interchange a and 5, the 
 second member becomes 
 
 hia^ — b2ai= — (ai&2 — «2^i) •*• (^i<^'2) = — (f^i^a) • 
 
 (b) The interchange of two subscripts. 
 
 {a^h^ =ai62 — ^^2^1' If the subscripts are interchanged, the 
 second member becomes 
 
 a^hx — a-Jy^ = — («i?>2 — «2^i) •*• (^2^1) = — (^^2) • 
 
 2d. For determinants of the third order. 
 
 (a) The interchange of two letters. 
 
 (tti^aCs) = («i^2)c3— {(^\bz)<^2+ («2^3)<^i- In this, if we inter- 
 change a and &, the proposition is obvious from the first part 
 of the demonstration, (a). 
 
 We have therefore to show that the proposition holds for b 
 and c. We have, 5, 
 
 {aib.^ {a^c^) — {a^b^) {axG,^)^ a.{o-ih<^8) - 
 
 In this expression, interchanging b and c, the first member 
 becomes {a-^c^) (^^s) — {(^'2<^z) (<^hh)' Since a^ remains un- 
 changed, (aiCg&s) = — (r^i^oCs). 
 
 (b) The interchange of two subscripts. 
 
 (oi 60 C3) = (tti 62)^3— (^1^3) <^2+ («2 ^3)^1- (I')- If the sub- 
 scripts 2 and 3 are interchanged, the second member becomes 
 {aibs)c2— (ai&2)c3+ («3&2)ci. Since {cisbo) = — («2^3)? 1st, 
 (6) , the second number of (L) becomes 
 
 — (ai&2)c3+ {aibs)Co— (a.2bs)ci 
 
 In the same manner it may be shown that the interchange of 
 any other two subscripts in (L) changes the sign of the second 
 \ member, .*. the proposition. 
 
10 THEORY OF DETERMINANTS. 
 
 3d. For determinants of the fourth order. 
 
 (a) The interchange of two letters. 
 
 (aiftgCa^O = {a^hcz)d^— {a^hc^d^-^ {aih^c^d^— (a2&3<J4)^i (I*)- 
 From 2d, (a) , the proposition is obvious for an interchange 
 of the first three letters. To show that the proposition holds 
 for c and d, we have, 6, 
 
 The interchange of c and d transforms the minuend into sub- 
 trahend, and the subtrahend into minuend, in the first member. 
 Hence, as (ag^s) remains unchanged, {aib2d3C4) = — {aiboC^di) . 
 
 (b) The interchange of two subscripts. 
 
 {a^b^c^d^) = {aAcs) d^ — (aib2C^) d^ + (a^hc^) d^ — {aihc^ d^. (M) . 
 
 In this, if we interchange the subscripts 2 and 3? the second 
 member of (M) becomes 
 
 (ai 63 C2) (^4 — («! 63 C4) (^2 + (oti 62 C4) do, + (% &2 C4) di. 
 
 Now, by 2d, (6), {aib^c^) = — (aib^Cs) ; and (ag&aQ) = — (ot2&3C4). 
 Hence the second member of (M) may be written 
 
 — (tti &2 C3) d^ + (ai 62 C4) ds — («! &3C4) ^2 + (02 &3 C4) di, 
 and therefore (ai 6302^4) = — (ai b2C3d^) . 
 
 In a similar manner the proposition may be established for 
 the interchange of any other two subscripts. 
 
 It is obvious that two consecutive interchanges will leave the 
 determinant unaltered either in sign or magnitude. Notice 
 that an interchange of two letters corresponds to a uniform 
 change in the order of succession of the unknown quantities in 
 the original set of equations. Also, that an interchange of two 
 subscripts corresponds to changing the order of the equations. 
 
 10. Applying the proposition of the preceding article to the 
 values of cc, y^ z, and t, obtained in 8, we have 
 
 __ (mi 62^3 0^4) , _{0im2C^di) ^ _(ai?)2^3<^4). . _ ((^ib^c^m.i) 
 "■ (% 62^3^4) ' {dibiCsd^) ' ~ (ai&2C3C?4)' ~ {ctibzC^d^)' 
 
NOTATION. 11 
 
 Notice that the common denominator in these" values is the 
 determinant of the fourth order, formed from the coefficients of 
 the unknown quantities. Also, that the numerator of the value 
 of X is obtained by changing the a of the denominator into m. 
 The numerator of the value of y is likewise obtained by chang- 
 ing the b of the denominator into m, and that the numerators 
 of the values of z and t are similarly obtained b}^ changing the 
 c and d into m respectively. 
 
 Notation. 
 
 11. We have seen that a determinant of the second order 
 contains 2^=4 quantities, a determinant of the third order 
 3^ = 9 quantities, and a determinant of the fourth order 4^ = 16 
 quantities. It is customary to employ the notation introduced 
 b}- Cayley, and write these determinants so that the quantities 
 (called elements) entering into the determinant appear arranged 
 in the form of a square, with a vertical line on each side. 
 
 Thus (aib2)= cti^i ; (aibzCs)^ aybyCi and (aj 62^3(^4)= aibiCidi 
 
 02^2 ^<2^2C2 a2b2C2d2 
 
 a^b^Cid^ 
 
 Other forms of notation are also | cti 62 1 for (aj 62) ; | ai 62 Cg | 
 for (0162^3); |ai&2C3<^4l ^ov {aib2Csdi). 
 
 There are still others to be described later. In Cayley's 
 notation the elements are so arranged that, regarded as coeffi- 
 cients of the unknowns in the original set of equations, they 
 occur in rows and columns in the regular order in which they 
 are found in these original equations. Further, comparing the 
 expansions with the square arrangement, we notice that each 
 term contains one, and only one^ element from each row and 
 cohimn, and that there is no other element from the same row 
 and column in the same term. Hence, as already exemplified, 
 there can be only 2, 3, or 4 elements in each term, according 
 as the determinant is of the second, third, or fourth order. 
 It will be noticed that the quantities occurring in the abbreviated 
 
12 
 
 THEony or determinants. 
 
 forms (oihoCo), (ciiho^), \a1b.2 0^(1^1,, etc., are those found in one 
 of the diagonals in the square arrangement, viz., the diagonal 
 extending from the upper left-hand corner to the lower right- 
 hand corner. This diagonal is called the principal diagonal. 
 Similarly, that diagonal extending from the lower left-hand 
 corner to the upper right-hand corner is the secondary diagonal. 
 Any line parallel to these (principal or secondary) is a minor 
 diagonal. Any of the expansions heretofore given show that 
 the product of the elements of the principal diagonal is a posi- 
 tive term of the determinant. This term being composed of 
 the elements of the principal diagonal, is called the principal 
 term. The other terms can be formed from the principal term 
 by making all the possible permutations of the subscripts and 
 prefixing the proper sign to each permutation (5 and footnote ; 
 also 6). 
 
 Observe that the order of the letters in the abbreviated forms 
 of notation is the order of the columns in the square arrange- 
 ment, and that the order of the subscripts gives the order of 
 the rows. Thus, [ai^gCsl means the determinant whose first 
 column consists of a's, second column of &'s, and third column 
 of c's, and that the subscript of each letter in the first row is 1, 
 and in the second each letter has the subscript 2? and in the 
 third each letter has the subscript 3. 
 
 Illustrations are : 
 
 
 
 
 
 
 |a3&2C4| = 
 
 «3 h C3 
 a.2 b., Co 
 04 64 C4 
 
 ' 
 
 
 
 
 1036405(^1! = 
 
 V 
 
 «3 63 C3 d^ 
 a^h^c^d^ 
 
 «'6 h C5 d^ 
 Oi hi Ci di 
 
 ; («i 
 
 C2bs) = 
 
 Oi Ci 61 
 a2C2&2 
 
 030363 
 
 \a,h,c^d,\ = 
 
 Oy b^ c^ 
 a. b, c, 
 
 a,b,c. 
 
 dr 
 
 d. 
 d. 
 
 • 
 
 ' 
 
 ♦• 
 
^*^iVE:^s\\ 
 
 i £lLIFORN\^ 
 NOTATION. 
 
 13 
 
 The expansion of determinants of the second and third orders. 
 
 12. Though we have alread}' given the expansion of deter- 
 minants of the second and third order several times, it will be 
 useful here to compare these expansions with the square 
 arrangement once more. Also, we are now prepared for a 
 convenient mnemonic rule for the expansion of a determinant 
 of the third order, to be given in 15. 
 
 13. Since I «i&i =ai&2 — ^2^15 it is obvious that the expan- 
 
 I Ct2 &2 
 
 sion of a determinant of the second order is obtained by taking 
 the product of the elements of the principal diagonal and the 
 product of the elements in the secondary diagonal, and sub- 
 tracting the second product from the first. 
 
 14. We have repeatedly shown that 
 
 Ci. 
 
 ttl 61 tl 
 
 = 
 
 ai6i 
 
 C3- 
 
 a,b. 
 
 C2 + 
 
 aa^a 
 
 Osf&gCa 
 
 
 0^2 62 
 
 
 a^h 
 
 
 «3&3 
 
 agftgCg 
 
 
 
 
 
 
 
 From this it appears that a determinant of the third order can 
 be decomposed into determinants of the second order, each 
 multiplied by the elements in order of the last column, begin- 
 ning with the last element. Since any column may be made 
 the last, 9, the assertion just made amounts to saying that 
 a determinant of the third order may be expressed in terms 
 of determinants of the second order and the elements of any 
 column. 
 
 The reader will readily see how the determinant factors of 
 the expansion in the present article are obtained from the 
 original determinant. For example, the ciifactor of Cg is ob- 
 tained by striking out the row and column in which c^ is found, 
 and regarding what is left as a determinant of the second order. 
 Thus, I 
 
 (h ^3 ?3 
 
14 
 
 THEORY OF DETERMINANTS. 
 
 15. The following convenient rule for the complete expan- 
 sion of a determinant of the third order is indicated in the 
 accompanying diagram, and is described as follows : — 
 
 The terms composed of ele- 
 ments of the principal diagonal 
 and of the minor diagonals 
 parallel to it are positive, while 
 those formed of elements in 
 the secondary- diagonal and the 
 minor diagonals parallel to it 
 are negative. The elements 
 pierced by the double lines 
 compose the positive terms. 
 The elements pierced by tlie 
 single lines similarly consti- 
 tute the negative terms. In accordance with these directions, 
 the expansion of 
 
 6^2 0.2 C^ 
 «3 h ^2, 
 
 is ai^a^'sH- «2^3Ci + da^iCg — ag^gCj — aa&iCg — aih^C2. 
 
 This is identical with the expansion already obtained in 5, 
 it should be. 
 
 as 
 
 EXAMPLES. 
 
 1 . Find the values of : 
 
 4 -G 
 
 5 3 
 
 25 18 
 49 75 
 
 10 -6 
 8 -3 
 
 a b 
 b a 
 
 7 1 
 5 
 
 2. Write in determinant form : 
 
 a-\- b b 
 a -\-b a 
 
 3 
 4 
 
 1 
 
 a -b b\; 
 b —c c \ 
 1 
 a 
 
 7 ; 5 : 16 ; —13;Xiy — xy', 3a — 7b ; (^ — bd; ; Sgh — xy. 
 
 b a 
 
 3 -1 
 
 1 2 
 
 (Suggestion :— 7 = 3x2 — (Ix— 1) 
 
 Numberless 
 
 other forms could, of course, be given for the same quantity.) 
 
EXAMPLES. 
 
 15 
 
 3. Without passing from the determinant notation, show 
 what relation exists between 
 
 (9.) 
 
 a, b. 
 
 and 
 
 b^ai 
 62 ^2 
 
 . Also 
 
 X y 
 m n 
 
 and 
 
 m n 
 X y 
 
 • 
 
 Compare 
 
 a b 
 c d 
 
 and 
 
 a c 
 b d 
 
 
 Also < 
 
 3ompa 
 
 re 
 
 a b 
 G d 
 
 5 ' 
 
 3a 
 
 c 
 
 36 
 d 
 
 , and 
 
 3a 
 3c 
 
 b 
 d 
 
 5. Write the expansion of the following determinants : 
 {a^h); (a^bj,); \a^biG„,\; {a^b^CQl; (ag 65 Cj) ; 162^3^1 
 
 6. Find the values of : 
 
 12 3 
 
 5 
 
 4 7 8 
 
 5 
 
 a 
 
 ? 
 
 0a 
 
 1 
 
 a c 
 
 5 
 
 a 6 c 
 
 4 5 6 
 
 
 3 6 
 
 
 6 
 
 
 b c 
 
 
 6 6 
 
 
 b c a 
 
 7 8 9 
 
 
 5 9 
 
 
 c 
 
 
 6 
 
 
 c a 
 
 
 cab 
 
 7. Compare 
 
 Also compare 
 
 Also compare 
 
 8. State the probable theorems exemplified by the results 
 Ex. 7. 
 
 a c 
 d f 
 g Ok 
 
 and 
 
 a 
 d 
 9 
 
 . 
 h k 
 
 a mb c 
 d me f 
 g mh k 
 
 and 
 
 a b c 
 d e f 
 g h k 
 
 tti 61 Ci 
 
 a.2 62 C2 
 
 ttg 63 Cg 
 
 
 and 
 
 a 
 6 
 c 
 
 1^2 «3 
 
 1 ^2 h 
 
 1 ^2 C3 
 
 9. Find the value of x in the equations : 
 
 (1) 
 
 (3) 
 
 X 2 
 
 = 
 
 3 4 
 
 ; (2) 
 
 1 -1 
 
 
 5 6 
 
 
 4 1 
 3 -2 
 2 1 
 
 = 
 
 1 
 
 1 
 
 1 
 
 a 
 
 X 
 
 c 
 
 6 
 
 6 
 
 X 
 
 = 0; (4) 
 
 X a a 
 
 = — 
 
 6 6 a; 
 
 a X a 
 
 
 6 a; 6 
 
 a a X 
 
 
 a; 6 6 
 
16 
 
 THEORY OF DETERMINANTS. 
 
 10. Find the complete expansion of 
 
 = I a^ 6< c„ cZ^I . (6, equation (R) , et seq.) 
 
 a. 
 
 K 
 
 Crd^ 
 
 a, 
 
 b, 
 
 Ct d, 
 
 «» 
 
 K 
 
 dn^n 
 
 a. 
 
 \ 
 
 Cod, 
 
 11. Wiite in determinant form, square notation: 
 
 (1) bfg-\- eid -\- lick — hfd — ecg — bik. 
 
 (2 ) Till Tig rg — my % rg + mg n^ r^ — m^ Ui r^ + mg Wj ?'2 — WgTigri. 
 
 (3) Sxyz-a^-f — ^. 
 
 12. Employ 9 to compare the following : 
 
 a b c 
 d e f 
 g h'k 
 
 and 
 
 d e f 
 g h ^k 
 a b c 
 
 'i 
 
 m n 
 p q r 
 
 s t u 
 
 and 
 
 n m 
 r q p 
 u t s 
 
 
 m n 
 
 and 
 
 m n 
 
 , 
 
 
 
 p q r 
 s t u 
 
 
 
 r I 
 
 U i 
 
 q 
 f t 
 
 
 
 13. Expand the following in terms of determinants of the 
 second order and the elements of any column (14). Verify 
 the results by making use of the rule in 15 : 
 
 «! 6i Ci 
 
 1 
 
 002 2/2 ^2 
 
 5 
 
 a^ 64 C4 
 
 CL2 bo C2 
 
 
 ^*3 2/3 '^h 
 
 
 as b. C3 
 
 «3 h Cs 
 
 
 Xi y^ m^ 
 
 
 C/6 ^6 Cs 
 
 14. Count the inversions of order in 
 
 (a) 13 5426 7 
 
 (6) 2 3 6 1457 
 
 (c) 6 3 5 4 17 2 
 
 (d) 789653421 
 
 (e) 987654321 
 
CHAPTER II. 
 
 GENERAL PROPERTIES OF DETERMINANTS. 
 "^ Notation and Definition. 
 
 16. The investigations of the preceding chapter have revealed 
 the fact that a determinant of the second, third, or fourth order 
 is a function of 2^, 3^, or 4^ quantities respectively, and have also 
 established a uniform law of formation for these functions. In 
 order therefore to investigate the properties of Determinants in 
 general, we have but to consider a function of ?i^ quantities 
 whose law of formation is given in the following definition. 
 
 17. Definition. — A Determinant is always a function of n^ 
 quantities. These quantities, called elements, being arranged 
 in the fonn of a square consisting of n rows, and thus also of 
 n columns, n quantities in each row and in each column, the 
 determinant of these n^ quantities is the sum of the terms 
 formed as follows : * Each term is the product of n elements, 
 so chosen that there is one element from each row and one 
 from each column, — but two elements from the same row or 
 column must never occur in any one term. The sign-factor of 
 each term is (— 1)^+', in which p is the number of inver- / 
 sions of order t of the rows, and i is the number of inversions > 
 of order of the columns, from which the elements composing 
 the term have been chosen. 
 
 Note. — Each term being composed of n factors, the deter- 
 minant is said to be of the nth order or degree. 
 
 * 22 et seq. will show that the law of formation given in this definition 
 is the same as that already observed in determinants of the 2d, 3d, and 
 4th orders (3 to 6 inclusive). 
 
 t 5, footnote on inversions of order. 
 
18 
 
 THEORY OF DETERMINANTS; 
 
 18. To expand 
 
 a 
 
 b c 
 
 d 
 
 ef 
 
 mn o\ 
 
 by the definition, we may select any 
 
 row, as, for instance, the second row, and using each element* 
 of that row in turn, according to the directions given, we shall 
 form all the terms of the determinant. For the first term, then, 
 taking d as the first element, we see that we can take b and o 
 
 iy, h c 
 
 -M n 
 
 for the other factors of a term, and no more, since we have 
 then chosen one element from each row and one from each 
 column, and no two elements are from the same row or column. 
 We now have the term dbo. To form another term containing 
 d, we can evidentlj' take n and c, giving the term dnc^ which as 
 before contains an element from each row and column, and no 
 two elements are from the same row or column. No other 
 terms containing d can be formed. The terms containing e are 
 in the same way eao and mec ; the diagram will sufficiently 
 explain the manner of obtaining these terms. 
 
 a '1}^ c 
 
 m i^ 
 The terms containing / are likewise naf and fbm. 
 
 a 1) ^ 
 m n 6 
 
 To fix the signs of these terms, we will write under each 
 term the numbers giving the rows and the numbers giving the 
 
 * There is a difference in the nomenclature. What we have called 
 elements some authors call constituents, and an element is a term. 
 
GENEKAL PllOPERTIES OF DETERMINANTS. 
 
 19 
 
 columns from which the elements have been taken, and opposite 
 eacli series the number of inversions. Thus : 
 
 dho dnc eao mec naf fhm 
 
 Rows 213-1 231-2 213-1 321-3 312-2 213-1 
 Columns 123-0 123-0 213-1 123-0 213-1 321-3 
 The sum of the inversions of order in rows and columns of the 
 first term is unity; .-. (— 1)^= — 1, and dho is negative. In 
 dnc the sum of inversions of order in rows and columns is 2 ; 
 .-.(—1)^=1, and dnc is positive. Similarly for the other 
 terms. Affecting the terms with their proper signs, 
 
 a h c 
 d e f 
 m n o 
 Scholium. - 
 
 dho + dnc -f eao — mec — naf -\- fhm. 
 
 This illustration is inserted only to give the 
 reader a clear idea of the meaning of the definition, and not 
 '|*> because we really employ the definition in the practical expan- 
 sion of determinants. In fact, the great beauty of the deter- 
 minant notation is that we are able to conduct most of our 
 investigations with the help of determinants without requiring 
 the expansions at all. In case it becomes necessary to expand 
 a determinant, we have several excellent methods to be given 
 later. One method for the expansion of a determinant of the 
 third order has been given already (15) . 
 
 19. In accordance with the notation already exemplified in 
 Chapter I., a determinant of the nth order is written 
 tti hi Ci ... li 
 
 O2 0-2 C2 ... I2 
 
 a^ hs C3 ... Is 
 
 <.ln\, or to 
 
 a„ 6« c„ ... l„ 
 This form is shortened to (ajdaCa ... /„) or \a1h2C3 
 2 ± cti&aCg ..*. l^. In each of these shortened forms those ele- 
 ments occur which occupy the principal diagonal* in the square 
 arrangement. The form S ± ai^a^s ... ?„ is suggestive of the 
 manner in which the function is formed. The 5 ± stands for 
 
 * 11. 
 
20 
 
 THEORY OF DETERMINANTS. 
 
 the sum of all the terms that can be formed from the pnncipal 
 term b}' permuting the subscripts and prefixing the proper sign 
 to each. (23.) 
 
 Another and very convenient notation is obtained by employ- 
 ing a single letter affected with two subscripts ; the first sub- 
 script giving the row, and the second subscript the column, in 
 which the element occurs. Thus : 
 
 ail ^12 ^% ••• %» 
 
 OE'21 tl22 <^23 ••• ^2w 
 a^i a^2 <^33 ••• <^3» 
 
 This form may, like the first, be shortened to | «ii «22 ••• ct„„|, 
 (ciii a22 ^33 • • • ^nn) , or 2 ± «!! «22 «33 • • • «„»!• It may also be still 
 further abbreviated to | ai^ | . A modification of this notation, 
 with the two subscripts, consists in omitting the letter alt^j 
 gether, and writing the determinant thus : 
 
 (1,1) (1,2) (1,3) ... (l,n) 
 (^2,1) (2,2) (2,3) ... (2,n) 
 (3,1) (3,2) (3,3) ... (3,71) 
 
 (n,l) (n,2) (n,3) 
 or, finally, /I 2 3 
 
 ,. (n,n) 
 
 or 
 
 11 
 
 12 13 . 
 
 . In 
 
 21 
 
 22 23 . 
 
 . 2n 
 
 31 
 
 32 33 . 
 
 . 3 71 
 
 nl 
 
 n2 7i3 . 
 
 .. 7171 
 
 ', /I 2 3 ... 7iY 
 1,1 2 3 ... n) 
 
 These last three forms are called the iimbral notation. 
 
 20. The following corollaries flow from the definition in 17. 
 They are obvious upon a moment's reflection. 
 
 Cor. I. — The principal term is always positive. 
 
 Cor. II. — If each element of a row or of a column is zero, 
 the determinant vanishes. 
 
 ^^ General Properties. 
 
 21. Theorem. — If in a series of integers tvhicJi are 
 
 aJI 
 
 different^ any two are interchanged^ the others remaining undis- 
 turbed, the number of inversions of order is thereby increased or 
 diminished by an odd number* 
 
GENERAL PROPERTIES OF DETERMINANTS. 21 
 
 Let the series of integers be Ae Bf (7, in wliich A is used to 
 denote the series ay/c ... preceding e, B denotes the series hgl ... 
 between e and /, and G the series following f. 
 
 In the first place, it is evident that if any two adjacent 
 integers are interchanged, the number of inversions of order 
 is thereby increased or diminished by unity. For let vm be 
 any two adjacent integers in a series. ^If we write mv, we 
 introduce one inversion of order if m >?;. Or, if m<v, we 
 have lost an inversion. Now, since this change cannot affect 
 the rest of the series, we have increased or diminished the total 
 number of inversions in the series by unity. 
 
 Again, in order to interchange e in Ae Bf (7, with / separated 
 from e by 7t, intervening elements, we may first interchange e 
 with the elements to the right in regular succession k-\-l times ; 
 I this brings e into the place at first occupied by /. Then, in 
 order to transfer / to the place formerly occupied by e, we have 
 to pass / over k elements to the left. Altogether, we have 
 changed the *number of inversions of order from odd to even, 
 or from even to odd, 2k-\-l (an odd number) of times. Hence 
 the proposition. 
 
 22. Theorem. — The number of terms in a determinant of 
 the nth order is 1 -2 '2> • ... n = n\ 
 
 The simplest wa}' to form the terms of a determinant accord- 
 ing to the definition, is to choose the elements from the columns 
 in order ; that is, the first element of a term from the first 
 column, the second element from the second column, etc. 
 Choosing the elements in this wa}', we may take the first ele- 
 ment of a term from the first column and third row, say, the 
 next element from the second column and any row except the 
 thirds the next element from the third column and any row 
 except those already selected, and so on, until all the columns 
 and rows have been drawn upon. The numbers of the rows 
 from which the elements are chosen will constitute a permu- 
 tation of the numbers 1, 2, 3, ... n, and at is obvious that 
 we can therefore select the elements to form a term in as" 
 
22 THEORY OF DETERMINANTS. 
 
 many different ways as there are permutations of the first n 
 numbers, that is n ! There are accordingly n ! different terms. 
 
 23. Cor. I. — The terms of a detenninant | aidgCg ... ?„ I 
 may all be obtained by keeping the letters in alphabetical order 
 (^.e., choosing the elements for each term from the columns in 
 order), making all the possible permutations of the subscripts, 
 and prefixing the sign + or — to each permutation, according 
 as the number of inversions of order is even or odd. Since 
 the expansion of a determinant in accordance with the definition 
 would also be obtained by keeping the rows in order, and 
 choosing the elements from the columns in all possible ways. 
 all the terms of | ai 62C3 ... Z„ | can be formed by permuting the 
 letters, keeping the subscripts in order, and prefixing the sign 
 + or — to each permutation, according as the number of in- 
 versions of the letters is even or odd. 
 
 24. Cor. II. — Similarly, the terms of | a-^^ \ can be formed 
 by making all the jjossible permutations of the first set of sub- 
 scripts and keeping the second set in order ; or the terms may 
 be obtained by making all the possible permutations of the 
 second set and leavinsr the first set in order. 
 
 Illustrations : To expand 
 
 , we may write the permu- 
 
 ctj hi Ci 
 
 (^2 ^2 ^2 
 % O3 C3 
 
 tations of the subscripts in a column, and indicate the number 
 
 of inversions of order in each by a figure placed at the right ; 
 
 or we may write the permutations of the letters in the same 
 
 way. Thus : 
 
 1 2 3 ... ah c ... 
 
 1 3 2 ... 1 ach ...I 
 3 1 2 ... 2 6 a c ... 1 
 3 2 1 ... 3 5 ca ... 2 
 
 2 3 1 ... 2 ca 6 ... 2 
 2 1 3 ... 1 c & a ... 3 
 
 The two expansions are accordingly 
 
 a^hoC^s — ai&gCo + ciz^iCi — ct^hoCi + aa^^aCi — cTo^iPsj 
 
GENERAL PROPERTIES OF DETERMINANTS. 
 
 23 
 
 To expand |aiia22«33| according to Cor. II, we have simply 
 to write the elements for each term with one set of subscripts 
 in order ; thus, 
 
 ai(X2<^35 %<^2«3^ «l«2%1 aia2%1 ftl<^2«35 «1«2<^3) 
 
 and then for every term, according as we choose from columns 
 or rows in order, write one permutation of the numbers 1, 2, 3, 
 before or after the subscripts already written, obtaining 
 
 ClllClooCtsS — ^'^11^32^^23 "T" ^31^2^23 — ^31^22^13 "T" ^21^%2^''13 <^21^12^33 
 
 or 
 
 %1^22^'33 — %lC'^23^32 — <^12<^21^33 ~\~ ^n<^23%l i ^13^% ^32 ^13^^22 '^Sl* 
 
 25. Theorem. — In any determinant, if the rows in order 
 are made the columns in order, the determinant is unchanged. 
 
 The theorem is an obvious consequence of 23 and 24. The 
 following proof is based directly upon the definition. Consider 
 the determinants A and A', which differ only by making the 
 rows of the one the columns of the other. Every term of A 
 contains an element from each row and column of A ; hence it 
 contains an element from each column and row of A', and is 
 therefore, disregarding the sign, also a term of A'. Similarly, 
 ever}^ term of A' must be a term of A. We have now to show 
 that the signs of corresponding terms are alike. Let the num- 
 bers of the rows and columns for a term of A be 
 
 a? y-i P-) Ti 0-9 ••• for the rows ; 
 
 r, t, a, s, 771, ... for the columns. 
 Then, by hypothesis, the numbers of the rows and columns of 
 the corresponding term from A' will be 
 
 r, t, a, s, m, ... for the rows ; 
 
 «5 y? i^? T-i ^? ••• for the columns. 
 The two terms obviously have the same sign. Hence the 
 proposition. 
 
 Illustrations : 
 a^i ai2 cii3 ^14 
 
 ^21 ^22 ^23 ^24 
 %1 ^32 ^33 <^34 
 a^i €1^2 ^43 ^'44 
 
 «U «21 «31.<*41 
 <^12 ^22 %2 ^'42 
 
 ai3 a2s a^g a^ 
 
 ^14 <^^24 <^34 ^44 
 
 (Xi bi Ci di 
 
 2 ^2 2 *^ 
 
 % ^3 ^3 ^4 
 a^ 64 C4 di 
 
 cfci tta as a4 
 bi 62 i>3 h 
 
 Ci C2 C3 C4 
 
 di do ds d^ 
 
24 THEORY OF DETERMINANTS. 
 
 26. Theorem. — In any determinant the number of positive 
 terms equals the 7iumher of negative terms. - 
 
 By 23 all the terms of a determinant can be formed by keep- 
 ing the letters in order, and making all the possible permuta- 
 tions of the subscripts (or 24, case of the double subscripts, 
 by keeping one set in order and permuting the other set) . AVe 
 
 n\ n\ 
 
 have to show, therefore, that -^ permutations are even* and -^ 
 
 are odd.* Let x and y be the number of even and odd per- 
 mutations respectively; then x-\-y = n\ If we interchange 
 any two subscripts in each of the aj even permutations and in 
 each of the y odd permutations, the even permutations become 
 odd and the odd even. Since by the interchange of two sub- 
 scripts we could only reproduce permutations all different from 
 each other, and already found in the original set of permuta- 
 tions, it follows that x = y. 
 
 27. Theorem. — If two parallel lines (rows or columns) of 
 a determinant are interchanged^ the sign of the determinant is 
 changed, but its numerical value is unchanged. 
 
 Let A be the given determinant and A' the same determinant 
 after the A:th and rth rows have been interchanged. Then 
 -A= A'. 
 
 Let J^ = ± Adj^Bm^C be a term of A, in which A, B, and C 
 denote the product of elements from all the rows and columns 
 except the cZth column and A;th row, and the mill column and 
 rth row. Then T (disregarding the sign) is also a term of A', 
 for it contains an element from each row and column of A'. 
 Now T, regarded as a term of A', contains exactly the same 
 inversions of the columns as it does when regarded as a term 
 of A ; but the number of inversions in T, as to rows, when 
 considered as a term of A', is an odd number, more or less, 
 than when considered as a term of A. For, in writing the 
 numbers of the rows, to determine the inversions, we write 
 
 * This language, of course, signifies permutations in wliicli the number 
 of inversions of order is even or odd respectively. 
 
GENERAL PIIOPEKTIES OF DETERMINANTS. 
 
 25 
 
 them just as we would for A, except that Aj and r will have 
 changed places (dj, being found in the rth row, and m^ in the 
 Tcih. row of A') . Thus every term of A is found with the oppo- 
 site sign in A', .•. — A = A'. By 25 the proposition must be 
 equally true for an interchange of two columns. 
 
 Illustrations 
 
 ttj hi Ci 
 ao 69 Co 
 
 «l ^1 Ci c?i 
 02 &2 Co d2 
 «3 ^3 C3 (k 
 a^ 64 C4 ^4 
 
 — [«! 62 C3 + ^2 63 Ci + % ^1 Co 
 
 f^lXX 
 
 — ttg ^2 Ci — tto 61 Co — a^ 63 Co] =( 7 5 
 
 ai bi Ci f7i 
 ^2 &2 ^2 c^_> 
 a4 64 C4 CZ4 
 
 % ^3 ^3 ^^3 
 
 ttj bi Ci cli 
 
 «3 ^3 C3 <i3 
 a4 &4 C4 ^4 
 a2 ^2 ^-2 ^2 
 
 «! (^1 Cj bi 
 
 a^ ^4 C4 64 
 
 (ai^gCsfy = — (a^biCsd^) = {a^b^Cid^) = — (a2 53C4di) = (o3Z>2C4di). 
 
 28. Cor. — If two parallel lines of a determinant are iden- 
 tical, the determinant A^anishes. 
 
 For, by the proposition, if the two identical rows or columns 
 are interchanged, the sign of the determinant is changed. But 
 the interchange of two identical lines cannot affect the deter- 
 minant. Therefore 
 
 A= -A, 
 2A = 0, or A = 0. 
 
 Illustrations 
 
 a b*c 
 
 = aec^ 
 
 def 
 
 
 a b^^ 
 
 «! Oj c 
 
 
 
 a.2 b.2 Co c?2 
 
 
 ag 60 C2 c?2 
 
 •v 
 
 0-4 64 c 
 
 ,^4 
 
 
 ai 
 
 a.2 
 
 a^ 
 
 «4 
 
 h 
 
 h 
 
 h 
 
 h 
 
 Ci 
 
 C2 
 
 C2 
 
 C4 
 
 ^1 
 
 d^d^ 
 
 C^4 
 
 aec — dbc — a6/= 0. 
 
 0. 
 
 (Ctj 60 Co C?4) = 0. 
 
 tti 62 «3 C?4 I = 0. 
 
 29. If in a series of integers, 
 
 I /, a, d, c, Z, m, w. 
 
26 
 
 THEORY OF DETERMINANTS. 
 
 the first is passed over all the others in succession to become the 
 last, the others remaining undisturbed, thus, 
 
 a, d, c, Z, m, n, /, 
 
 the numbers are said to have been cyclically interchanged. It 
 
 is obvious that a cyclical permutation of n given numbers 
 
 can always be effected by n — 1 interchanges of two adjacent 
 
 W numbers. Accordingly, a permutation containing an odd or 
 
 {^ even number of inversions still contains, after a cyclical inter- 
 
 ^ change, an odd or even number of inversions if n is odd ; if n 
 
 is even, however, a pepoiitation containing an_o_dd or even 
 
 number of inversions will, after a cyclical interchange, contain 
 
 an even or odd number of inversions respectively. 
 
 From a given permutation of n integers any other permuta- 
 tion can be obtained by cyclical interchanges. Thus, from 
 
 faclcegb 
 we get c a gfd b e 
 
 as follows : — cfadegb 
 
 c afd e g b 
 
 c a gfd e b 
 
 c a g fd b e 
 The groups in which the cyclical interchanges take place are, 
 of course, fade, fa, fdeg, /, d, eb. 
 
 30. The previous article (or 27) establishes the following 
 theorem : 
 
 Theorem. — If in a determinant A any row or column be 
 passed over k rows or columns in succession, and the resulting 
 determinant be denoted by A', then 
 
 A=(-1)'=A'. 
 
 Illustrations : 
 
 ^iViZiti 
 
 = 
 
 ^3 Vs ^3 ^3 
 
 = 
 
 ^3 h 2/3 2=3 
 
 = — 
 
 x^ ?i 2/1 z, 
 
 x^y^z^h 
 
 
 XlVlZih 
 
 
 iCi fi yi Zi 
 
 
 X2 /a y-i Z2 
 
 aJ3 2/3 »3 ^3 
 
 
 X2y2%2t2 
 
 
 X2 h 2/2 22 
 
 
 3^4^4 2/4^4 
 
 x,y,z,t. 
 
 
 x^y^z^t^ 
 
 
 x^ U 2/4 z^ 
 
 
 X'ihy3^3 
 
 ^0 2/l V2lVs\= — \ Xi ^2 VsWo\ = —\ViX2 2/3 ^^0 | = 1 «1 2/2 f^3%\' 
 
GENERAL PKOPEKTIES OF DETERMINANTS. 27 
 
 EXAMPLES. 
 
 1. The student who has not done the examples at the end 
 of the first chapter may attend to them before proceeding to 
 the following. 
 
 2. What terms of | aib2G^di \ contain ftgf^s? 
 
 . 3. Write the terms of (a^i 2/2 ^3 2:4 Q that contain ^1 2/4 w^s. 
 
 4. Show that in a determinant of the nth order only two 
 terms can have (n — 2) elements in common, and that these 
 terms have opposite signs. 
 
 5. What is the sign-factor of the term containing the ele- 
 ments in the secondary diagonal of a determinant of the nth 
 order ? 
 
 6. Show that the sign of a term is independent of the 
 arrangement of the elements composing it. 
 
 7. Show that the sign of a determinant is not changed by 
 an}^ interchanges of rows and columns that leave the same 
 elements in the principal diagonal, whatever the final arrange- 
 ment of the elements in this diagonal. 
 
 Syg. If a^p ayy a^a • • • ot-m be the final arrangement sought, 
 a^p can be brought into the first place by 2 (^ — 1 ) interchanges 
 of two rows and columns, etc. 
 
 8. A corollary from 30 is : Any element (X,;^ can be trans- 
 ferred to the first place b}' making the ith row and A;th column 
 the first row and column, and then multiplying the determinant 
 by (-1)*+*. 
 
 31. Theorem. — If every element of any line (row or column) \y 
 is multiplied by any number, the determinant is multiplied by 
 that number. 
 
 Since every term of the determinant contains one element, 
 and only one, from the line mentioned in the theorem, the truth 
 of the proposition is evident. 
 
28 
 
 THEORY OF DETERMINANTS. 
 
 Illustrations 
 
 a^r hyV CiT 
 
 = r 
 
 ai 61 Ci 
 
 = 
 
 «! ftjr Ci 
 
 = r2 
 
 «>,, 
 ^ &1C1 
 
 ^2 h C2 
 
 
 a2 h C2 
 
 
 (12 &2^ ^2 
 
 
 Clg 63 Cg 
 
 
 a-g 6g Cg 
 
 
 dg 63^ Cg 
 
 
 <^'2 , 
 
 
 
 yhC-I 
 
 
 
 «3 , 
 
 
 
 
 
 
 
 7 h c. 
 
 A = 
 
 6c a (T 
 ca h b^ 
 ah c c^ 
 
 then a&cA = 
 
 abc a^ a? 
 abc h^ b^ 
 abc & <? 
 
 •. A = 
 
 1 OL' a" . 
 
 1 62 53 
 
 1 C^C^ 
 
 Let the student show that 
 
 bed a a^a? 
 
 =2: 
 
 cda b b^ W 
 
 
 dab c c^ c^ 
 
 
 abc d d' d' 
 
 
 1 a^ a^ «* 
 
 1 62 53 J4 
 
 1 c2 c^ c^ 
 1 d^ d^ d^ 
 
 32. Cor. I. — Changing the signs of all the elements of any 
 row or column changes the sign of the determinant ; for it is 
 equivalent to multiplying the determinant by — 1. 
 
 33. CoR. II. — If two rows or two columns differ only by a 
 constant factor, the determinant vanishes. For we may divide 
 each element by the constant factor, and write this factor as a 
 multiplier before the determinant. Then the determinant van- 
 ishes by 28. 
 
 Illustratioi 
 
 IS : 
 
 i 
 
 
 al a** 
 
 = a" 
 
 al Z 
 
 =0. 
 
 ba a-+i 
 
 
 h0L a 
 
 - 
 
 C a^ a~ + 2 
 
 
 eV a? 
 
 
 1 5 7 
 
 2 10 6 
 345 0) 
 
 1 1 7 
 
 2 2 6 
 33 9 
 
 = 0. 
 
 34. Theorem. — If each element of any line * of a determinant 
 is a binomial^ the determinant equals the sum of two determinants; 
 the first of which is obtained from the given -determinant by sub- 
 stituting for the binomial elements the first terms of the binomialSy- 
 and the second determinant is obtained from the given determi' 
 
 * Since it has been shown (25) that what is true of the rows of a 
 determinant holds for the columns, it will only be necessary hereafter to 
 state a proposition with reference to either rows or columns. 
 
GENERAL PROPERTIES OF DETERMINANTS. 
 
 29 
 
 nant by substituting for the binomial elements the second terms of 
 the binomials. 
 
 By the definition ever}' term of the determinant must contain 
 one of the binomial elements. 
 
 Let (m-\-n) b g h k ,.. I 
 
 be one of the terms of the given determinant ; this may be 
 written m b g h k ... l-{-n b g h k ... I. 
 
 Now the first term of this sum is a term of the original deter- 
 minant, with m written for m-i-n, and the second term is a 
 term of the original determinant, with n written for m.-fn. It 
 is obvious that a similar statement applies to every term of the 
 given determinant ; hence the proposition. 
 
 Illustrations : 
 
 «! + «! h Ci 
 
 = 
 
 < 
 
 2l bi Ci 
 
 + 
 
 tti bi Ci 
 
 . 
 
 Cl2 + a2 &2 C2 
 
 
 a2 h C2 
 
 
 0-2 &2 ^2 
 
 
 a3 + «3 ^3 C3 
 
 
 % h Cs 
 
 
 ag 63 C3 
 
 
 Xi — yi mi Ui 
 
 = 
 
 Xi mi 7?i 
 
 — 
 
 2/1 «h n^ 
 
 0^2-2/2 ^2 ^2 
 
 
 X2 m-2 «2 
 
 
 2/2 '^2 no 
 
 a^-2/3 W3 Wg 
 
 
 
 Xs mg 7 
 
 ^3 
 
 
 
 2/3 'nis 
 
 nsl 
 
 35. The preceding theorem is evidently" capable of extension. 
 The same reasoning applies to a determinant any line of which 
 is composed of polynomial elements, or, again, in which each 
 element of every line is a polynomial. That is to say : If each 
 element of any row is a polynomial of q terms, each element of 
 another row a polynomial of r terms, each element of another 
 row a polynomial of s terms, etc., the given determiyiant is the 
 sum of sxq X r... determinants. Thus : 
 
 a-\-b -\- c m — n 
 d+e-f o+p 
 g — h-\-k q — r 
 
 t 
 u 
 
 V 
 
 
 1 
 
 i 
 
 7 
 
 m 
 
 Q 
 
 — n t 
 4-i) u 
 
 — r V 
 
 -f- 
 
 
 b m — n 
 
 e -\-p 
 
 -h q — r 
 
 t 
 
 u 
 
 V 
 
 c m — n t 
 
 —f o+p u 
 
 k q — r V 
 
 — 
 
 a m t 
 d u 
 g q V 
 
 + 
 
 a — n t 
 d p u 
 g — r V 
 
 + 
 
 b m t 
 
 e u 
 
 — hqv 
 
 
 b — n t + 
 e p u 
 — h — r V 
 
 A 
 
 1 n 
 
 
 i t 
 
 u 
 
 V 
 
 H 
 
 h 
 
 
 c — n 
 
 -f P 
 k-r 
 
 t 
 u 
 
 V 
 
 
 
 
30 
 
 THEORY OF DETERMINANTS. 
 
 36. Reciprocally, If q determinants differ from each other 
 only in a single line, the sum of these determinants is a single 
 determinant, derived from any of the given determinants by sub- 
 stituting for the elements of the line which is different in each 
 of the q determinants, the sum of the corresponding elements of 
 the q determinants. 
 
 Illustrations : 
 
 ab c 
 
 + 
 
 ab c 
 
 + 
 
 mn 
 
 = 
 
 
 a b c \ 
 
 def 
 
 
 X y z 
 
 
 ab c 
 
 
 d-{-x — m e-\-y — n f-\-z — o 
 
 ghk 
 
 
 g hk 
 
 
 ghk 
 
 
 g h k 
 
 The student may show that 
 
 
 «! «i 61 Ci 
 
 — 
 
 tti bi Ci 
 
 = 0. 
 
 
 0-2 «'2 h C2 
 
 
 a^ 62 C2 
 
 
 
 ttg bs C3 
 
 
 ^3 ^3 C3 ag 
 
 
 
 
 
 
 
 a^ &4 
 
 C4 
 
 
 a^ b^ C4 a^ 
 
 
 37. Theorem. — A determinant remains unchanged if the 
 elements of any line be increased or diminished by equal multiples 
 of the corresponding elements of ayiy parallel line. 
 
 We are to show that 
 
 tti 61 Ci 
 
 tta 62 ^2 
 ^3 ^3 ^3 
 
 «i ^1 Ci±qiai±q2bi±...±li 
 a2 62 C2±qia2±q2b2±...±L 
 <X3 &3 Cs±qias±q2b3±...±l 
 
 d, ... h 
 
 4 ••• ^s 
 
 a„ 6„ c,,±qia^±q2b^±...±l, d, ... /, 
 
 Calling the first determinant A, and the second A', we have, 35, 
 A' = 
 
 a, bi Ci d, . 
 
 (/2 60 Cg (^2 • 
 
 
 ±qi 
 
 ± 
 
 «i ^1 «i ^^1 • 
 
 0^2 ^2 «2 C^2 • 
 
 ./2 
 
 aa 62 ?2 ^2 • 
 
 
 (^n ^n ^n <^n • 
 
 ai 61 61 di . 
 
 a2 ?>2 ^2 ^^2 • 
 
 
 »n K «n <^n^- 
 
 
 
 «n K K dn . 
 
 . L 
 
 
 
 «n ^n ^« dn . 
 
 .. I, 
 
 :t<?2 
 
 Whence, since all the determinants of this series, except ILe 
 first, A^anish, A = A'. 
 
GENERAL PROPERTIES OF DETERMINANTS. 
 
 31 
 
 This theorem is of great importance in simplifying and ex- 
 panding determinants. Thus : 
 
 lab-\-c = la 
 1 b c + a 1 b 
 1 c a + b 1 c 
 
 a a+3 a+6 
 a + 1 a + 4 a-\-7 
 a + 2 a + 5 a-f-8 
 
 a + 6 + c | = (a4-6 + c) 
 a + 6 +c 
 
 1 a 1 
 1 & 1 
 1 c 1 
 
 = 13 (a 
 
 
 1) 3 (a + 4) 3 (a + 7) 
 
 1 a + 4 a-f7 
 
 2 a-l-5 a + 8 
 
 = 0, 
 
 The second determinant is obtained by adding the second 
 and third rows to the first row. 7 
 
 = -16. 
 
 1 
 
 1 
 
 1 
 
 1 
 
 = 
 
 -11 1 1 
 
 1- 
 
 -1 
 
 1 
 
 1 
 
 
 /0%2 2 
 
 1 
 
 1 
 
 -1 
 
 1 
 
 
 //O 2X2 
 
 1 
 
 1 
 
 1 
 
 -1 
 
 
 J 02 2 
 
 =r — 
 
 2 2 
 
 = —8 
 
 1 1 
 
 
 202 
 
 
 1 1 
 
 
 2 20 
 
 
 1 1 
 
 The second determinant is obtained by adding the first row to 
 each of the others. 
 
 The third determinant is obtained from the second by ob- 
 serving that as all the elements, except one, of the first column 
 of that determinant are zeros, all the terms vanish that do not 
 contain (—1). 
 
 7 
 
 13 
 3 
 
 11 
 15 
 9 
 
 4 
 10 
 6 
 
 7 
 
 11 
 
 13 
 
 15 
 
 1 
 
 3 
 
 -240. 
 
 4 
 
 = 3 
 
 7 
 
 -10 
 
 -10 
 
 = 3 
 
 10 10 
 
 10 
 
 
 13 
 
 -24 
 
 -16 
 
 
 24 16 
 
 2 
 
 
 1 
 
 
 
 
 
 
 
 = 30(16-24) = 
 
 The third determinant is obtained from the second by subtract- 
 ing three times the first column from the second column, and 
 twice the first column from the third. 
 
 / 
 
 Minor Determinants. 
 
 38. If in a determinant any number of rows and the same 
 number of columns are suppressed, the determinant consisting 
 of the remaining elements (their relative positions being undis- 
 turbed) is called a minor of the given determinant. 
 
 If one row and one column are suppressed, the result is a 
 principal minor, or a Jirst minor ; if two rows and two columns 
 
32 THEORY OF DETERMINAMTS. 
 
 have been erased, a second minor ; and so on. The elements 
 common to the suppressed rows and columns also form a deter- 
 minant called the complementary of the minor, formed from the 
 rows and columns that were left undisturbed in the original 
 determinant. 
 
 Thus, 
 
 «3 ^3 
 
 and 
 
 C4 ^4 
 
 are complementary minors of 
 
 I «! 62 C3 c?4 I ; also I «i 62 1 and | Cg (^4 1, or &2 and | a^ C3 CZ4 1 , are 
 complementary minors of | aj 62 C3 CZ4 1 . | c^ d^ \ and | a., 64 e^ \ are 
 complementary minors of \ai'b2C.id^e^\. In general, if the 
 determinant is of the nth order, two complementar}" minors will 
 be of the rth and {n — r)th orders respectively. A determinant 
 
 n^ (71 ~ IV 
 of the 71 th order has rr first minors, — ^^ L second minors, 
 
 etc. 
 
 Since we usually denote a determinant by A, it is convenient 
 to denote the minor obtained by suppressing the row and 
 column of a^ by A^g ; that obtained by suppressing the row 
 and column of d^ by A^^^, etc. 
 
 Similarly, a second minor, obtained by suppressing the rows 
 and columns of b^ and c^, is denoted by Abh.Cr ; and so on. 
 
 Equally efficient notations are : D ) and D} ^^ -. for the 
 
 minor obtained by suppressing the Zth row and mth column, 
 and the minor obtained by suppressing the Zth, wth, and tih. 
 rows, the mth, rth, and h\h columns respectively of | %,i | . 
 
 39. Since, by definition, every term of a determinant con- 
 tains one, and onl^^ one element from any line, the determinant 
 must be a linear homogenous function of the elements of any 
 one row or column. Thus : 
 
 I tti 62 C3 ... ^^ 1 = «! J.1 + a2-42 + a^A^ -\ f- a,,A,, 
 
 = «! A + \B, + ci Ci + ••• + /i A 
 
 = ci Ci + C2 a, + C3 (73 + .-. + c„ a 
 
 in which ^1, A2...A,, ; Ci, (72...C„ ; etc., denote functions of the 
 elements found in the rows and columns outside of the particu- 
 
GENERAL PEOPERTTES OF DETERMINANTS. 33 
 
 lar line, in terms of which the development is given. In the 
 next article we shall find the values of these functions. Since 
 we may regard the determinant as a function of r^ independent 
 quantities, each of the coefficients A^^ A^^ ..., may be obtained 
 by differentiating | aiftgCs-.-Z^ | with reference to the quantity 
 whose coefficient is desired. Introducing this concept, the 
 equations above written become respectively 
 
 I ai62C3...?„ 
 
 , cZA , cZA , cZA , 
 cZttj da^ da^ 
 
 cZA 
 
 •• + «,-— 
 
 da^ 
 
 cZA , , cZA , cZA , 
 dai "Oi dci 
 
 J dA 
 
 ^ d\ , clA ^ cZA , 
 dci cZcg (Zc3 
 
 
 This notation is often employed. 
 
 /^^^s^xI'heorem. — The coefficient of any element in the expan- 
 sion of a determinant is the first minor obtained by suppressing 
 the row and column to which the elemeyit belongs. This 7riinor 
 is taken with the + sign, if the sum of the row and column 
 numbers, to which the element belongs, is even; if this sum is 
 odd, the minor has the — sign. 
 
 Consider the determinant A = 2 ± «i &2 ^3 . . . In-, and suppose 
 A to be written, 
 
 A = ai^1l + «2^2 + «3^3+ ••• +«nA- (1) 
 
 We can collect all the terms of A that contain ai, and write 
 this element as a factor of the polynomial that results ; we can 
 do the same for a^', a^, and so on, for each element of the first 
 column. These pol^'nomials are ^1, A2, A^, etc. Now A^ must 
 be composed of all the terms of A that contain no elements 
 from the first row or column ; hence A^ can be obtained from 
 2 ± «! &2 C3 ... Z^ by considering ai as fixed, and making all the 
 possible permutations of the subscripts of the remaining letters, 
 i.e., by multiplying aj by IS ± &2 ^3 ••• hv 
 
 Hence, ^1 = A„j, the minor. obtained by suppressing the first 
 row and first column of A. 
 
34 
 
 THEORY OF DETERMINANTS. 
 
 V' 
 
 Now, we can bring 0^2 into the first place by one interchange 
 of rows : we then have A = — 2 ± ag 61 Cg ... Z„. Employing the 
 same reasoning as before, A2 must be obtained by multiplying 
 ^2 by — 5 ± &i Cg ... Z„ ; whence A2 = — Aa^. Again, a^ can be 
 brought into the first place by two interchanges of two rows ; 
 whence A^ = A^g, and so on. Finally, a^ can be brought into 
 the first place by n — 1 interchanges of two rows ; hence, as 
 before, A =(-!)""' Aa«. 
 
 Substituting these values of Ai, Az, etc., in (1), 
 
 A = tti Aa^ - a2^a, + %Aa3 h ( - 1 ) ""^ a^ Aa«. 
 
 Since the columns may be made rows, it is evident that 
 A = aiAa,-&iA6^+CiAc, + (-l)"-*ZiAv 
 
 It remains to be shown that the proposition holds for an ele- 
 ment not in the first row or column, that is, A^^ = ( — 1)*'^'' Aaa, 
 for the coefficient of the element in the iih row and Jcth. column. 
 We may transfer the ith row to the first place by i— 1 inter- 
 changes of two rows, and the Jcth column may likewise be made 
 the first by A: — 1 interchanges of two columns. The element 
 under consideration is now in the first place. Calling the trans- 
 formed determinant A', we have 
 
 A = (-l)*+*-2A', orA = (-l)^+*A'. 
 Whence ^,, = ( - 1 ^ +* Aa^, . 
 
 41. CoR. I. — A determinant can be developed in terms of 
 the elements of any line and their principal minors. The signs 
 are alternately + and — ; and the fii'st term is + or — , accord- 
 ing as the number of the line is odd or even. 
 
 Illustrations : 
 
 
 
 
 
 
 
 tti bi Ci f?i 
 
 = «! 
 
 62 C2 d2 
 
 -a^ 
 
 h Ci c^i -\- as 
 
 b, c, d, 
 
 -a^ 
 
 61 Ci dy 
 
 O2 ^2 ^2 d.2 
 
 
 h Cg dg 
 
 
 h Cg dg 
 
 h C2 di 
 
 
 62 C2 (?2 
 
 «3 ^3 Cg ^3 
 
 
 64 C4 d^ 
 
 
 64 C4 ^4 
 
 64 C4 d4 
 
 
 h<hd^ 
 
 ^4 64 C4 6^4 
 
 
 
 
 
 
 
 
 = — 61 1 as Cg ^4 I 4-&2 1 «i Cg d^ I — ftglai Cg d^ 1 -f ?>4 1 «l C2 dgj 
 
 = Ci I a2 63 (^4 1 — C2 1 ai &3 ^4 1 4-C3|ai &2 1?4 1 — C4 | «i ^2 ^3 1 
 
 = - «4| ^ C2 (^3 I +^4 1 % C2 C?3 I -C4|ai hi ^3 | -f-Cf4| tti &2 C3I 
 
GENERAL PROPERTIES OF DETERMINANTS. 
 
 35 
 
 42. 41 obviously gives a ready way of expanding any deter- 
 minant.* For we may express the given determinant in terms 
 of the elements of any line and their principal minors ; these 
 minors will be determinants of the (n — 1 ) th order. By a 
 second application of 41, each of the minors in the first expan- 
 sion ma^' be expressed in terms of the elements of any line and 
 their principal minors, which minors will be of the (?i — 2)th 
 order. So by successive application of 41, an}' determinant 
 may be expressed in terms of determinants of the second order ; 
 and these latter, being binomials, can be at once written out. 
 Thus : 
 
 1 2 3 
 
 = 13 4 
 
 -2 
 
 2 3+3 
 
 2 3 
 
 2 3 4 
 
 4 5 
 
 
 4 5 
 
 3 4 
 
 3 4 5 
 
 
 
 
 
 1(15-16) -2(10-12) +3(8-9) =0. 
 
 \aib2Csd^\^=ai\b2Csd^\—a2\biCsdi\-^as\biC2di\—a^\biC2ds\ 
 
 = ai[b2\csd^\-b^\c2d^\+bi\c2ds\2-a2lbi\Csd^\-b^\cidi\-\-bi\cids\2 
 
 +asibi\c2di\—b2\cid^\-\-b^\cid2\']—a^lb^\c2ds\—b2\cid-^\-\-b.i\cid2\^ 
 
 = aiboC^d^ — aj 62^4^3 — «i 6302(^4 + ai 6304^2 + aibiC2ds — a^b^c^di 
 
 — a2 &i C3 C?4 + (^2 ^1 C4 c?3 + 
 
 -\-a^biC2d^ — a.;ib^G^d2 — 
 
 — a^b^Cid^ + 0146103^2 + aib2Cid^ — a^biC^d^ — a^b^Cid2 + aJy^Cidi 
 
 43. As another corollary from 40, it is evident that if all 
 the elements of any row of a determinant except one are 
 zeros, the determinant equals this element into its corresponding 
 minor, taken with the proper sign. Thus, if the element is in 
 the ith row and A:th column, i.e., a,.;^, then A = ( — l)'+*a,4Aa,.^. 
 
 Illustrations : 
 
 0-2 
 1 1 
 1 
 
 5 643 
 
 = -2 
 
 6 4 3 
 1 1 1 
 1 2 1 
 
 = -2 
 
 2-0 
 3 111 
 
 
 3 
 
 = 2 
 
 -2 
 
 -3 
 
 1 
 
 
 
 1 
 
 
 
 * Compare 15. 
 
 t Compare 6. 
 
36 
 
 THEORY OF DETERMINANTS. 
 
 The student may establish the following : 
 
 «! 
 
 h 
 
 c,d. 
 
 
 
 h 
 
 c,d. 
 
 
 
 
 
 CsCh 
 
 
 
 
 
 d. 
 
 ctiboCgd^. 
 
 di 
 
 Ca ^2 
 &3 Cg d^ 
 . — ^^b^c^d^ 
 
 a^hc^di. 
 
 44. 
 
 From the last two examples it appears that if all the 
 elements on one side of either diagonal are zeros, the determinant 
 reduces to a single term, viz., the term composed of the elements 
 in the diagonal which contains no zero elements. 
 
 2 1 - 
 
 -4 -3 
 6 5- 
 
 7 
 8 
 9 
 
 = 2 
 
 1 1 
 
 2 3 
 
 3 5 
 
 7 
 8 
 9 
 
 • 
 
 Show that 
 
 
 
 a 
 
 
 
 . /3 
 J' (3' 
 
 'I" 
 
 = 
 
 1 
 al3y 
 
 EXAMPLES. 
 
 1. Show that the following determinant vanishes : 
 
 2. 
 
 1 1 1 
 
 a' (By /S' ya y' a/B 
 a"fty /S'V" y"^^ 
 This can be readily established by multiplying the columns 
 by (Sy, ya, a^, respectively, and then dividing the first row by 
 aySy. A similar reduction can he effected, in general, tchenever it 
 is desired to reduce a determinant to one in which the elements 
 of one line are units. 
 
 4. Find the expansion of A 
 
 4 2 5 10 
 116 3 
 73 5 
 025 8 
 
 We notice that 20 is the L.C.M. of the elements in the first 
 row; hence, multiplying the columns in order by 5, 10, 4, 2, 
 there results 
 
 A = 
 
 1 
 
 5.10.4-2 
 
 20 20 20 20 
 
 
 5 10 24 6 
 
 — 
 
 35 30 10 
 
 
 20 20 16 
 
 
 1 
 
 1 
 
 1 
 
 11 
 
 5 
 
 10 24 6 1 
 
 7 
 
 6 
 
 
 
 2 
 
 
 
 5 
 
 5 
 
 4 
 
GENERAL PEOPERTIES OF DETERMINANTS. 
 
 37 
 
 / 
 
 Now, subtracting four times the first row from the fourth 
 row, two times ,the first row from the third row, and six times 
 the first row from the second row, the last determinant becomes 
 
 1 1 
 
 1 1 
 
 
 
 _ 
 
 -1 4 
 
 18 
 
 . 1 
 
 71 -14 18 
 
 1 4 
 
 18 
 
 
 5 4 
 
 -2 
 
 -3 6 -2 
 
 5 4 
 
 -2 
 
 
 -4 1 
 
 1 
 
 1 
 
 4 1 
 
 1 
 
 
 
 
 = —6 
 
 71 
 
 — 7 
 
 = -6 
 
 64 -7 =-3 
 
 
 
 
 
 -1 
 
 1 
 
 
 1 
 
 5. 
 
 also 
 
 (Xj (X2 (X3 
 61 &2 &3 
 C.i C2 C3 
 
 1 a a' 
 IIS 13' 
 
 1 
 
 
 1 1 1 
 
 aia20 
 
 C2 C3 
 
 ^3 * 
 < 
 
 ^2 (Xs &i «! 0-3 62 ^1 Cf' 
 
 1 1 
 
 1 ttg 62 Cj a2 ^3 ^1 
 1 asbiC2 a2^C3 
 
 2&3 
 2C3 
 
 = (/3-7) (y-a) (a-;8) 
 
 A vanishes if a = /S, or j8 = y, or a = y ; hence a — ^, /? — y, 
 and a — y must be f aotors of A. Now the product of the three 
 differences is a function of the third degree in a, ^, y ; so is A ; 
 hence the product of the three differences can differ from A only 
 by a constant factor. Comparing the term j3y^ (the principal 
 term), we see the factor mentioned is + 1. 
 
 7. Show that 
 1111 =_(^_^)(,_8)(^_„)(^_S)(,_^)(^_8). 
 
 a /3 y 8 
 a' /32 y2 §2 
 a^ ^3 ^3 33 
 
 Notice that Examples 6 and 7 give in determinant form the 
 product of the differences of the roots of an equation whose 
 roots are a, ^, y, ... 
 
 8. Expand 
 
 8 7 2 
 
 20 
 
 ; also 
 
 3 1 4 
 
 7 
 
 
 5 11 
 
 
 
 
 8 10 
 
 6 
 
 
 1 a 
 
 -a 1 
 (S -y 
 
 7 
 
 1 
 
38 
 
 THEORY OF DETERMINANTS. 
 
 Expand the first determinant in terms of the elements of the 
 third row and their principal minors, since two of these elements 
 are zero ; then observe that two elements in a row of each of 
 the resulting determinants are unity ; hence, each determinant 
 can be readily reduced to one of the next lower order by 35 
 and 42. «^ >> + 
 
 : V d^ + b^e^ + c'f-2 bcefr- 2 cafd - 2 abde. 
 
 c d 
 
 ; also 
 
 d c 
 
 
 a b 
 
 
 b -a 
 
 
 P 
 
 y, 
 
 y' 
 
 -P' 
 
 1 
 
 a' 
 
 a' 
 
 1 
 
 1 o 
 -a 1 
 
 -y P 
 
 11. Establish the following identity, and express either 
 determinant as the product of four linear factors : 
 
 X y z 
 
 X z y 
 
 y z X 
 
 z y X 
 
 111 
 
 = 
 
 1 Oz'f 
 
 
 1 z'O x" 
 
 
 1 y^z'O 
 
 
 12. Simplify 
 
 13. Show that 
 
 ai-\-hi + ki a2 + ?h-hh cis-^h + h 1 
 
 Ci-j-Jh-\-h C2-{-hs-\-h Cs-hh-{-h 1 
 1 1 10 
 
 X y-^z + t 
 
 y z-\-t-{-x 
 
 z t + x-^y 
 
 t x-\-y-{-z 
 
 x-j-y z + t 
 
 y-\-z t + x 
 
 z-{-t x + y 
 
 t + x y-\-z 
 
 = 0. 
 
 14. Express as a single determinant: 
 
 (1) \a^b^Cs\-^\a2b^c,\-\a^biC,\. 
 
 (2) I Qq 62 Cs I — I ao 63 Cs I — I ai 63 Cj I + 1 tti 62 C5I . 
 
 15. (Xi+aa+tta (i2-\-^h-\-(^A «3+«4H-«i a4+^i+«2 
 6i+?>2+?>3 b2+h+b^ ^3+^+^ 64+^+^2 
 
 Cl+C2-f-C3 C24-C3+C4 C34-C4+C1 C4+C14-C2 
 d^-\-d2 + d^ C?2 + <^3 + <^4 C^3 + ^44-f?l C?4-f(Zi + d2 
 
 = 3 1 ciibiC^d^ |. 
 
GENERAL PROPERTIES OF DETERMINANTS. 
 
 39 
 
 16. 
 
 17. 
 
 sin^^ 
 
 6sin^ 1 cos^ 
 
 csin^ cos^ 1 
 
 cv'-{h^+c'-2bGGo^A). 
 
 a+h-\-nG (n — l)a {n — \)h 
 (n — l)c h-\-c-\-na (ti— 1)6 
 (w — l)c (n— l)a c-\-a-\-nh 
 
 18. {a+hy d" (? =2abc(a-{-b-hcy. 
 
 a' {b-^cy a' 
 
 b' b^ ' {c^ay 
 
 19. What is the coefficient of ^34 in |ai5 1? 
 
 20. From the first five rows of {dib^c^d^e^fQ {/y/ig] write all 
 the possible minors that can be formed, and their complemen- 
 taries. 
 
 How many minors, each a determinant of the hth. order, can 
 be formed from any k rows of | ai„ | ? 
 
 21. If each of the elements of any line is the sum of the cor- 
 responding elements of two or more parallel lines, multiplied 
 respectively by constant factors, the determinant vanishes. 
 
 22. Show that 
 
 tti 61 Ci 
 
 = 
 
 10 
 
 = 
 
 «i h Ci 2/1 
 
 = 
 
 «1 61 Ci Ui Vi . 
 
 «2 \ ^2 
 
 
 Xi «! 61 Ci 
 
 
 ^2 h C2 2/2 
 
 
 a.2 &2 ^'2 ^^2 '^2 
 
 «3 ^3 C3 
 
 
 072 ^2 ^2 ^2 
 
 
 «3 h C3 2/3 
 
 
 ttg ?>3 C3 W3 -^3 
 
 
 x^ as h C3 
 
 
 1 
 
 
 1 V4 
 
 
 
 
 
 
 
 1 
 
 From this example it appears that any determinant may be 
 expressed as a determinant of higher order by tvriting a zero 
 above every column, prefixing a 1 to the row of zeros thus formed, 
 and filling in the new column having 1 at the top with any n 
 finite quantities. 
 
 ,K 23. If in any determinant each element of the first row is 
 unity, and if each element of every other row is the sum of the 
 elements above and to the left of it in the preceding row, 
 commencing with the element directly above, the determinant 
 equals 1. 
 
40 
 
 THEORY OF DETERMINANTS. 
 
 24. Any determinant of order w, in which one element is 
 zero, is equal to the product of two factors, one oC which is a 
 determinant of the nth order, in which every other element of 
 the row and column containing the zero is unity. 
 
 25. If in any determinant the first element is zero, and if 
 each of the remaining elements in the first row and first column 
 is unity, the determinant is unchanged when each element of 
 the minor corresponding to the zero element is increased or 
 diminished by the same quantity. 
 
 26. A determinant of the nth. order is expressible as the 
 sum of n determinants, the first of which is obtained by chang- 
 ing into zero each element of any line except the first element, 
 the second by changing into zero the elements of the same line 
 except the second element, and so on. 
 
 27. If in two determinants A, A' of the nth order, the first 
 row of A is the last row of A', the second row of A the (n — l)th 
 row of A', the third row of A the (n— 2)th row of A', and so 
 
 on 
 
 then 
 
 A=(-l) 
 
 n (n-1) 
 
 2 A'. 
 
 28. If in two determinants A, A' of the nth order, the first 
 row of A when reversed is the last row of A', the second row 
 of A when reversed is the (n—l)th row of A', the third row of 
 A when reversed is the (w — 2)th row of A', etc. ; then A = A'. 
 
 45. Theokem. — If the elements of any line in a determinant 
 are respectively multiplied by the complementary minors taken 
 alternately plus and minus {i.e., the co-factors) of the correspond- 
 ing elements of any parallel line, the sum of the products is zero. 
 
 Consider the two determinants, 
 A 
 
 oti bi Ci 
 
 02 &2 C2 
 
 ... I, 
 ... k 
 
 and 
 
 A' = 
 
 ai bi Ci ... 
 ^2 Z>2 C2 ... 
 
 
 O'lc b„ Cj, 
 
 ... Ifi 
 
 a, 6, c, ... 
 
 Ik 
 
 % K % 
 
 ... Ip 
 
 «A h Ck ••• 
 
 ''k 
 
 ttn K C„ 
 
 ... t^ 
 
 CtnKCn ... 
 
 "rt 
 
GENERAL PROPERTIES OF DETERMINANTS. 41 
 
 where A' differs from A only in having the Jcth and pth rows 
 identical. Employing the notation of 39, and expanding in 
 terms of the elements of the pth row, 
 
 Comparing these two expansions, we observe that the second 
 may be obtained from the first by substituting for the elements 
 of the pth. row of A the elements of the Zcth row ; that is to 
 say, if the elements of the kth row of A are multiplied by the 
 co-factors of the corresponding elements of the pth row, the 
 result is A' ; since A' = 0, the proposition is established. 
 
 Illustrations : 
 
 If in (aib2Cs) = ai (b2C.]) — a2 (biC^) -{- a^ (biC2) we multiply 
 the elements of the second column respectively by the com- 
 plementary minors of «!, ag, as, there results 
 
 &i (&2C3 — bsC2) — 62 (^1^3 — &3C1) + &3 (^1^2 — &2C1) = 0. 
 Let the student prove the proposition, using a determinant 
 of the fourth order. 
 
 46. A determinant is said to be zero-axial if each element of 
 the principal diagonal is zero. Thus the following are zero- 
 axial determinants : 
 
 
 
 \ 
 
 Ci 
 
 » 
 
 61 Ci c?i 
 
 a2 
 
 
 
 C2 
 
 
 (X2 C2 C?2 
 
 as 
 
 h 
 
 
 
 
 04 ^4 C4 
 
 47. Theorem. — Any determinant may be decomposed into a 
 sum of zero-axial determinants : the first of these is obtained by 
 substituting zero for each element of the principal diagonal of the 
 given determinant ; the next ?i, by miiltiplying each element of 
 the principal diagonal by its complementary minor made zero- 
 
 n 
 axial; the next -(71— 1), by multiplying each product of pairs 
 
 of elements of the principal diagonal by its complementary minor 
 made zero-axial^ and so on. 
 
42 
 
 THEORY OF DETERMINANTS. 
 
 In A^"' = \a1b2C3... l,,\ change the elements of the principal 
 
 denote the resulting determi- 
 
 h 
 
 diagonal into zeros, and let aJ" 
 nant. Whence, 
 
 bi Ci 
 
 Ar^ 
 
 a^ 
 ttg bs 
 
 b^ c„ ... 
 
 Let aJ"""^^ denote the minor of Aq"^ obtained by suppressing 
 any one row of Aq" ; Aq"" denote the minor obtained by sup-- 
 pressing any two rows of Aq'^ ; and, in general, let Aq*'"'^ denote 
 the minor obtained by suppressing any i rows of Ao"\ Also 
 let C2 denote any product of the elements of the principal 
 diagonal of A^"^ taken 2 and 2 ; Cg any product of those elements 
 taken 3 and 3 ; and, in general, (7, any product of the elements 
 of the principal diagonal of A^**^ taken i and i. Now, Aq** 
 evidently contains all those terms of A^*"^ which involve no 
 element from the principal diagonal. CiAo"~ must be one of 
 those terms of the series which involve only a single element 
 from the principal diagonal of A^**^ ; consequently :S Cj Ao**"^^ 
 will be the sum of all the terms that contain only one element 
 from the principal diagonal of A^''^ Similarly, 1<C2'^1'^~^^ will 
 be the sum of all the terms that contain only two of those 
 elements. And, in general, SCfAj""'^ will be the sum of all 
 the terms containing i elements of the principal diagonal of 
 ^(n)^ Whence, 
 
 (n-2) 
 
 A("> = Ar + :s o, Ar''+ 2 c, Ar""+ :s c.a^o 
 
 (n-3) 
 
 + ...+2C,A 
 
 (n-i) 
 
 It is to be noticed that Aq^' = ; i.e., there is a break in the 
 series, — there being no term containing only 7i — l of the 
 elements in the principal diagonal. 
 
 Illustration 
 
 «i 2/1 2^1 
 
 ^2 2/2 ^2 
 ^3 2/3 ^3 
 
 2/1 ^1 
 
 X2 Z2 
 
 ^3 2/3 
 
 + Xi 
 
 Z2 
 
 2/3 
 
 +2/2 1 % 
 
 + z, 
 
 2/i|+«i2^2^3- 
 X2O I 
 
GENERAL PROPERTIES OF DETERMINANTS. 
 
 43 
 
 48. Theorem. — If each consecutive pair of elements in the first 
 row of a determinant A is taken with each pair of corresponding 
 elements of the other consecutive rows to form determinants of 
 the second degree^ and if these determinants of the second degree 
 are used in order as the elements of a new determinant A', then 
 A equals A' divided by the product of all the elements except the 
 first ayid last in the first row of A. 
 
 We are to show that 
 
 1 
 
 ai 6i Ci 
 
 ttg &2 ^2 
 «3 ^3 C3 
 
 ... ri Zi 
 ... r2 k 
 
 ... 7*3 k 
 
 ttn K Cn 
 
 ... r. In 
 
 h^c^d^...r^ 
 
 
 ai b, 
 
 aa &2 
 
 
 62 C2. 
 
 ... 
 
 ri Zi 
 ^2 Z2 
 
 
 «3 h 
 
 
 &1 Ci 
 
 ^3 C3 
 
 ... 
 
 ^3/3 
 
 
 ai 61 
 
 
 &1 Ci 
 
 ... 
 
 Tn In 
 
 calling the first determinant A, and the second A'. Multiplying 
 the first column of A by — 61, and the second column by %, 
 and adding, there results 
 
 -6iA = 
 
 
 a^ bi -j- cii 62 
 a^bi + a^b^ 
 
 Ci ... 
 
 C2 ... 
 
 Now, multiplying the second column by — Ci, and the third 
 column by &i, and adding, we have 
 
 61C1A 
 
 
 
 a2bi + aib2 — &2Ci + &iC2 
 a^bi-^a^bs —63^1 + ^1^3 
 
 ^3 h 
 
 —a^bi-\-a^bn -b^c^ + b^c^ c^...r^ l„ 
 Proceeding in a similar manner, we have, after (n—l) 
 transformations, 
 
 {-ir-'b,c,d,...kA 
 
 ... Zi 
 
 — a2bi + aib2 —b2Ci-\-biC2 — CgcZi + 01^3 ••• —^2^1 + ^1^2 k 
 
 — a^bi-haibs — ^s^i + ^Cg — Cgdj H-CicZg... — rgZi + riZg Z3 
 
 -a„6i+ai6„ -6„Ci-f&iC„ -c^di+CicZ„... -r„Zi + riZ„ Z« 
 
44 
 
 THEORY OF DETERMINANTS. 
 
 Now, applying 43, and dividing by ( — l)**~^6iCicZi ... ^i, 
 we have ^^ 
 
 |«i &2I l&i C2 
 l«i &3I 1^1 C3 
 
 7i ^3 
 
 -f- 6iCicZi...ri,^ 
 
 «! &n| 1^1 c„| ... |ri Z, 
 which establishes the x>roposition. 
 
 49. Since by the preceding proposition any determinant of 
 the njih order may be reduced to one of the (n— l)th order, we 
 have another means of simplifying any given determinant. 
 The proposition is especially advantageous in the reduction of 
 determinants whose elements are given numbers. Thus : 
 A = 
 
 12 3 4 
 
 _1 
 
 3 2 14 
 
 6 
 
 13 4 5 
 
 
 5 4 3 2 
 
 
 -4 -4 
 
 8 
 
 = -4 
 
 -1 -1 
 
 2 
 
 1 -1 
 
 -1 
 
 
 1 -1 
 
 -1 
 
 -6 -6 
 
 -6 
 
 
 1 1 
 
 1 
 
 24. 
 
 Here we can mentally reduce the determinants of the second 
 order obtained by combining the first pair of elements of the 
 first row with the corresponding elements of the other rows, 
 and obtain the elements of the first column of the new deter- 
 minant, thus : lx2-3x2 = -4; 1x3-1x2 = 1; 1x4 
 — 5x2 = — 6. For the elements of the second column we 
 have similarly : 2x1-2x3= — 4; 2x4-3x3 = — 1; 
 2x3 — 4x3 = — 6; and so on. 
 
 Let the student apply the proposition to show that 
 
 1 1 
 
 1 \-\-x 
 1 1 
 1 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 +2/ 
 
 1 
 
 1 
 
 1+^ 
 
 ^xyz', 
 
 also 
 
 10 
 4 
 3 
 
 7 
 
 17 
 8 
 8 
 20 
 
 13 
 6 
 1 
 17 
 
 = 124. 
 
 Also apply the proposition to show that 
 
 5 11 
 8 7 2 
 3 1 4 
 8 1 
 
 
 20 
 
 7 
 6 
 
 = 2188. 
 
EXAMPLES. 
 
 45 
 
 MISCELLANEOUS EXAMPLES. 
 
 1. Find the value of 
 
 2 4 3 1 4 3 
 -4 2-3 2 -1 2 
 
 5_1 6 2-1 5 
 1 11-2 -2-2 
 7_3_5 1 4 2 
 
 3 12-123 
 
 1 
 
 also of 
 
 12 22 14 17 
 16-4 7 1 - 
 10-3 -2 3 - 
 7 12 8 9 
 11 2 4 -8 
 24 6 6 3 
 
 20 
 -2 
 -2 
 11 
 1 
 4 
 
 10 
 15 
 8 
 6 
 9 
 22 
 
 
 2. Expand the following: 
 
 
 
 X ... a„ 
 -1 a; ic^ 3^3... a;"-ia„_i 
 
 0-1 ... a„_2 
 -1 ... a„_3 
 
 1 
 
 ai 62 ... 
 
 ^2 — ^1 ^3 ... 
 ttg — &2 64... 
 
 rx4 -63... 
 
 
 
 
 
 
 
 
 
 
 
 ... a, 
 ... -1 ao 
 
 
 a„ 0...- 
 a„+i ... 
 
 
 
 1 K+i 
 
 3. Show that 
 
 
 
 1 aa; + /^2/ + 9^2 
 10 7iaj4- 62/ +/2= 
 1 gra; -f /y 4- C2; 
 1 ?a;4-m2/ + w2; 
 a; 2/ 2; 1 h 
 
 
 1 
 
 
 
 X 
 
 
 1 
 
 
 y 
 
 
 
 
 1 
 z 
 
 ax -^ hy -\- gz -{- 1 
 hx -\-by -\-fz + m 
 gx + fy + cz + n 
 k 
 
 
 
 
 4. Write the complementaries of the following minors of 
 
 |ao 61 02^3 64/5]: 10264!; Ico/sl; I Co 6^4 62!; 1 61 63^4 1; [d^esfi] 
 
 5. What are the complementaries of 
 
 |ai2 (Xssl and | ai2 ass asg], in | aoi ai2 ass «34 «45 ^sel? 
 
 6. Show that 
 
 
 
 
 
 1 1 
 
 1 1+a 
 1 a+1 
 
 1 &H-1 6+a 
 1 c+1 c+a 
 
 1 
 
 1+6 
 
 a + 6 
 
 
 
 c + 6 
 
 1 
 1+c 
 a-\-c 
 6+c 
 
 
 
 = 23 
 
 1+a 1 1 
 1 1+6 1 
 1 1 1+c 
 
46 
 
 THEORY OF DETERMINANTS. 
 
 7. Prove that 
 
 \<h h\ K bs\ \(h &4|. 
 
 !«! C2I \ai C3I |ai C4I. 
 
 !«! d2\ [ai dal |ai d^]. 
 
 \aik\ \cii h\ \cii k\' 
 
 8. Employing the notation 
 
 /n\ ^ n{n-l)(n-2) ... (n-r + 1) 
 \rj~ rl . '■ 
 
 show that 
 
 tti a2 as a^ ... a^ 
 
 61 &2 h h '" K 
 Ci Cg C3 C4 ... c„ 
 di ^2 ^3 c?4 ... dn 
 
 1 
 
 ^n-2 
 «1 
 
 h h k k ... L^ 
 
 6.1 
 
 a, Z« 
 
 rn 
 
 ^ rx-\-2y\ /x+2tj+l\ /x+2y+2\ ^^^ /x+3y-l\ 
 Observe that ( ^ )-[ ^ j=( ^_, } 
 
 /x-{-y+l\ /a;+2/+2\ /aJ+2/+3\ 
 
 /aj+2/+2\ /x+y-^3\ ^x+y-{-A\ ^^^ /x+2y+l\ 
 
 The Product of Two Determinants. 
 
 V 50. If we note a determinant by K, and another by i, their 
 product P is evidently expressed by 
 
 L 
 
 The form of this product suggests the probability that the 
 
 product of two determinants may be expressed by writing the 
 
 factors as complementary minors of a determinant of higher 
 
 order, and filling in the vacant places due to one or both of 
 
GENERAL PROPERTIES OF DETERMINANTS. 
 
 47 
 
 the factors with zeros. Suppose, for example, that K is of the 
 third order, and L of the second ; then P would take the form : 
 
 «! bx Ci 
 
 «£ h C2 
 
 a4 ft 
 as ft 
 
 We now wish to discover if, when we fill in the vacant places 
 due to K or L with zeros, and thus make P a determinant of 
 the fifth order, P will still be the product of K and L. That 
 this is the fact will be shown in the next article. 
 
 51. Theorem. — The product of two determinants^ K and i, 
 of degree m and n, respectively^ is a determinant P, of degree 
 m-{-n, in which K and L are complementary minors^ so situated 
 that the principal diagonal of P is made up of the elements in 
 order of the principal diagonals of K and L ; the vacant places 
 in P, due to either K or i, are filled with zeros, and any mn 
 finite elements occupy the remaining places. 
 
 We have to show, for example, that 
 
 KxL 
 
 ai &i Ci 
 CI2 ^2 ^2 
 % ^3 C3 
 
 X 
 
 a4 ^4 74 
 as ft 75 
 a* ft 76 
 
 — 
 
 ai &i Ci 
 
 (I2 ^2 <^2 
 «3 ^3 C3 
 
 
 
 
 
 
 a^ 64 C4 
 
 «5 ^5 C5 
 
 ag b^ c. 
 
 a4 ft 74 
 as ft, 7s 
 a« ft 76 
 
 = p. 
 
 (1) 
 
 Developing P in terms of the elements of the fourth column 
 and their complementary minors, we have 
 
 P = 
 
 0-4 
 
 «! bi Ci 
 
 — as 
 
 ag 62 C2 
 
 
 ttg 63 C3 
 
 
 tts h C5 ft 75 
 
 
 «6 ^6 Co ft 7c 
 
 
 «! 61 Ci 
 
 a2 62 C2 
 
 as 63 C3 
 
 ^4 64 C4 ft 74 
 
 <^6 h ^6 ft 76 
 
 + a6 
 
 «! 
 
 ^'l 
 
 Ci 
 
 
 
 
 
 ^2 
 
 &2 
 
 C2 
 
 
 
 
 
 «3 
 
 &3 
 
 C3 
 
 
 
 
 
 ^4 
 
 h 
 
 C4 
 
 ft 
 
 74 
 
 «5 
 
 h 
 
 Co 
 
 ft 
 
 75 
 
 or 
 
 P = a4 Aa^ - a^ A^s + og Aae . 
 
 (2) 
 
48 
 
 THEORY OF DETERMINANTS. 
 
 But A, 
 
 ft 
 
 tti bi Ci 
 
 ttg 62 C2 
 
 ttg 63 C3 
 
 «6 ^6 Ce 76 
 
 -A 
 
 «1 
 
 h 
 
 Cl 
 
 
 
 a^ 
 
 h 
 
 C2 
 
 
 
 as 
 
 bs 
 
 C3 
 
 
 
 as 
 
 h 
 
 Co 
 
 75 
 
 = ftyel «i &2 C3 1 - ft 75 1 tti 62 C3 I = ^ I ft 75 
 
 I ft 76 
 In the same manner, we may show that 
 
 K 
 
 A 74 
 ft 76 
 
 also Aa^ = K 
 
 ft 74 
 ft 75 
 
 Substituting these values in (2) , we have 
 
 since the second factor is obviously L, expanded in terms of 
 the elements of the first column. 
 
 The method of proof here given is perfectly general, and is 
 applicable to determinants of an}^ order. Thus, if in (1) we 
 make y^ = y^z=zO, and 76=1, P takes the form considered 
 in 50. The student can readily make the application. 
 
 As another exercise, the student may show that 
 
 = -\ciif2C3d^\x\€,beg7 
 
 A= ai Ci di fi 
 ttg C2 (^2 /a 
 
 as C3 dg /g 
 
 ^4 C4 ^4 /4 
 
 65 65 9^5 
 b, e, g, 
 bj ej Qj 
 
 "What difference would it make in the result if tlie zeros in 
 the fifth, sixth, and seventh rows of A were replaced by any 
 finite elements? 
 
 52. Writing the product of | aj 63 Cg | and | iCi 2/2 2^3 1 , in accord- 
 ance with 51, 
 
 ai 61 ci - 1 = P, 
 ttg b.2 C2 0—1 
 
 ttg 6g Cg 0—1 
 
 x^ yi Zi 
 
 
 
 
 OOo 
 
GENERAL PKOPEKTIES OF DETEllMINANTS. 49 
 
 we have, by 37, 
 P = 
 
 
 
 
 
 
 
 aiX^ + a2yi-\-asZi &!«!+ &2 2/i + &3 2'i 
 
 a^x^+a^Vi + a^z^ biX2 -^ b2y2 -{- b^Zz 
 
 ttl^S + «22/3 + «32!3 b^Xs + 622/3 + b^Z^ 
 
 which, by 43, 
 
 ai4 + «2yi + «3^i ^ 
 
 «l»2 + «22/2 + "' ^ '^ 
 
 ai»3 + 42/3 + 
 
 -10 
 
 0-10 
 
 0-1 
 
 Cia^i + Cg 2/1 + ^3% xi 2/1 % 
 
 ^1^2 I <^22/2 "r <^3^2 **^2 2/2 % 
 
 CiiBs + c^^/s + Cs^Js a;'' 2/3 ^3 
 
 This result expresses the product of two determinants of the 
 third order as a determinant of the same order. We are thus 
 led to infer that the product of two determinants of any order 
 may be expressed at once as a determinant of the same order. 
 
 We now proceed to establish this important multiplication 
 theorem. , 
 
 53. Theorem. — The product of two determinants, A, A' 
 of the nth order is a determinant A" of the same order. Any 
 element a^s of £i" is obtained by midtiplying each element of the 
 rth row of A by the corresponding element of the sth row of A', 
 and adding the products.* 
 
 Before giving the general demonstration, it will be useful to 
 establish the proposition for the product of two determinants 
 of the third order, and note carefully the form of the result. 
 
 * Forming the product by columns, the statement is, of course : The 
 element in the rth column and sth row of A" is obtained by multiplying 
 each element in the rth column of A by the corresponding element in the 
 sth column of A', apd adding the products. 
 
50 
 
 THEOKY OF DETERMINANTS. 
 
 Put A = 
 
 «1 
 
 h 
 
 Cl 
 
 as 
 
 62 
 
 C2 
 
 as 
 
 63 
 
 C3 
 
 and A' = 
 
 o-i Pi 71 
 «» A 72 
 
 ^3 Ps 73 
 
 Apxjl3*ing the theorem, we have to show that 
 
 AA'=A" = 
 
 «iai + &iA + Ci7i «ia2 + &i/?2+Ci72 aia3 + 6i/83+Ci73 
 
 <^2ai + &2^1 + C2 7l «2a2 + &2/52 + C272 «2a3 + ^2i83 4-C273 
 
 %ai+63/?i+C37i a3a2+63/J2+C372 a3a3-i-^3A+C373 
 
 Since each element of A" is a trinomial, the determinant may 
 be decomposed into twent3'-seven determinants (35), the ele- 
 ments of which will be monomials. But of these twenty-seven 
 determinants only six do not vanish.* Those determinants 
 which do not vanish are formed by taking for the first column 
 a set of first terms from the first column of A", for the second 
 column a set of second terms from the second column of A", 
 for the third column a set of third terms from the third column ; 
 or, by taking a set of second terms from the first column of A", 
 a set of first terms from the second column of A", and a set of 
 third terms from the third column of A" ; and so on. That is 
 to say, exactly as many non-vanishing determinants can be 
 formed from A" as there are permutations of the numbers 1, 2, 
 3, i.e., 6. Hence 
 
 A'' = 
 
 ttiai 61^2 C173 
 «2ai ^2(^2 C273 
 «3<^i hPi C373 
 
 + 
 
 «!«! Ci72 5ift 
 
 a-iai C272 hfSs 
 a^a^ Csy2 h/Ss 
 
 + 
 
 61^1 ttittg Ci73 
 hPi «2a2 C273 
 
 bsPl ttgOa C373 
 
 
 + 
 
 5l/?i C172 dittg 
 
 &2^1 C272 «2a3 
 hPl C372 «3«3 
 
 + 
 
 Ci7i ^A CliOs 
 C271 69 A «2a3 
 
 C37I ^3^2 «3a3 
 
 + 
 
 Ci7i aia2 &ift 
 C271 a2a2 b.2(Ss 
 
 C371 ^302 Z^sft 
 
 
 = aiftys («i 2^263) - ai/?372 (aib,c.i) - a^f^iys {a.hc^) 
 
 + a3/5l72 («1&2C3) — a3/327l («1^2C3) + a2ft7l («1&2<^.0 
 
 (by 30 and 31) 
 
 * It is obvious that the determinants formed from sets of first terms 
 taken from the three columns of A", or those containing sets of first terms 
 from two columns, etc., must vanish. Similarly for determinants formied 
 from sets of second terms, and so on. 
 
GENERAL PEOPERTIES OF DETERMINANTS. 
 
 51 
 
 = (cLihCs) [ai/32 73+^2 /?3 71+^3 Ara— a3/?2 7i— a2A73 — aiftyg] 
 
 = («i^2C3)(aift73), 
 
 which establishes the proposition for the special case under 
 consideration. 
 
 In general, let 
 
 A = 
 
 «! 
 
 h 
 
 Ci 
 
 
 ^1 
 
 ^2 
 
 b. 
 
 C2 
 
 
 4 
 
 a. 
 
 bs 
 
 Cs 
 
 
 ^3 
 
 ttn 
 
 K 
 
 Cn 
 
 
 In 
 
 and A' 
 
 «! ft 7i ••• ■^i 
 
 «2 ft 72 ••• ^^2 
 
 "3 ft 73 ••• ■^S 
 
 ttn ft 7» ••• -^^ 
 
 Then the product AA' = A" 
 
 «!«! + ^ft + Ci7i + ... + hXl 
 
 «2ai-h&2ft + C2 7i + 
 
 <^3«l + ^3ft + C37i + 
 
 ..+?2Ai 
 ..+^3^1 
 
 Otn«l+ ^nft + C„7i+ ... + InK 
 «ia2+^ft + Cl72 + ... +^1^.2 
 
 a2a2 + 62ft + C272 4- 
 
 a3a2 + &3ft + C372 + 
 
 .. -j- /2A2 • 
 .. + ^3^2 . 
 
 «ian+&ift4-Ciy« + 
 
 «2an4-&2ft+C2 7„ + 
 «3an+&3ft+C37n + 
 
 
 a„a2 + 6„ft + c„72 4-.-.+^„ 2 ... «»an+^„ft+C„y„+... + Z„\ 
 
 • Now, A" may be decomposed into a sum of 7i~ determinants, 
 the elements of which are monomials. But it is obvious that 
 all those determinants whose columns are formed from sets of 
 first terms of the columns of A", or from sets of second terms, 
 etc., will vanish, as each will contain identical columns. In 
 fact, all those determinants into which A" is decomposed will 
 vanish that have not the first column formed from a set of kth. 
 terms from the first column of A'', the second column formed 
 from a set of rth terms from the second column of A", the 
 third column formed from a set of ^th terms from the third 
 column of A", and so on. Now, as many such non-vanishing 
 determinants can be formed as there are permutations of the 
 numbers 1, 2, 3 ... n ; that is, n ! Hence, A" is decomposable 
 into n ! determinants, of which the following Ar is the type : 
 
62 
 
 THEORY OF DETERMINANTS. 
 
 bipi kX2 aittg . 
 
 .. Ciy„ 
 
 &2^1 ^2-^2 Ci20.3 • 
 
 .. C27« 
 
 ^sA ^3^ (^3<h . 
 
 •. c^yn 
 
 Ar = 
 
 PnPl ^n^2 «»«3 ••• C„y, 
 
 But Ar = )8iA2a3 ... y„ I 6i?2a3 ••• c„|. 
 
 Now, the determinant factor of A^ is evidently A multiplied 
 by the sign-factor ( — 1)^, in which p is the number of inter- 
 changes of two columns which must be made in A to leave its 
 columns in the order which they have in Ar. Accordingly, 
 Ar=( — 1)^/81X203. ..y„A. But ( — iy/3iX2<^3"'yn is a term of 
 A', since the number of interchanges of two letters which must 
 be made in 01^273 ••• K to obtain the arrangement here given 
 is p. Accordingly, Ar equals a term of A' multiplied by A. 
 Thus each of the n\ determinants into which A" has been 
 decomposed is the product of A, and a term of A'. .*. A"= AA'. 
 
 Illustrations : 
 
 
 12 3 
 
 X 
 
 110 
 
 = 
 
 
 1110 
 
 
 3 2 11 
 
 
 
 3 2 1 
 
 
 10 1 
 
 
 
 12 
 
 
 2 110 
 
 
 0+2+0+0 3+4+0+3 
 
 0+1+1+0 3+2+1+0 
 
 0+0+2+0 9+0+2+1 
 
 0+0+1+0 0+0+1+2 
 
 1+0+0+3 2+2+0+0 
 
 l_|_0+0+0 2 + 1 + 1+0 
 
 34-0+0 + 1 6+0+2+0 
 
 0+0+0+2 0+0+1+0 
 
 2 10 4 4 
 
 =4 
 
 15 2 2 
 
 2 6 14 
 
 
 2 6 14 
 
 2 12 4 8 
 
 
 16 2 4 
 
 13 2 1 
 
 
 13 2 1 
 
 10 
 
 X 
 
 10 
 
 = 
 
 1 « a 
 
 
 1 a a 
 
 
 16^ 
 
 
 10 6/? 
 
 
 1 c y 
 
 
 1 C y 
 
 
 1 1 1 
 
 1 a^ + a^ ab -{- a/3 ac + ay 
 1 ab + afi b^ + (i^ bc-\-l3y 
 1 ac + ay &C + )8y (T + y" 
 
 54. Since, before multiplying two determinants together, we 
 may change the form of one or both factors, the product of two 
 determinants can be expressed in a variety of different forms. 
 As an illustration, the student may verify the following equa- 
 tions : 
 
GENERAL PBOPERTIES OF DETERMINANTS. 
 
 53 
 
 kxA 
 
 I a2'^02 
 
 
 aitti + aq/?! 
 ^itti + ^aA 
 
 (Xi ttj -|- 012 012 
 
 6iai + &2a2 
 
 a2a2 + 12/82 
 
 ttittg + ^2 A 
 6ia2 + &2/52 
 
 «2 A + &2/S2 
 
 «l/?l+«2A 
 5lft + 62/?2 
 
 EXAMPLES. 
 
 1. Show that one form of the product of 
 
 1 aa" 
 
 X 
 
 a^-a 1 
 
 is 
 
 1 6 &2 
 
 
 6^-6 1 
 
 
 1 c c^ 
 
 
 c'-c 1 
 
 
 
 W-ch + c" (? 
 
 2. One form of the product of 
 
 a+6 c c 
 a 6+c a 
 6 6 c+a 
 
 -i& 
 
 (? 
 
 a" 
 
 &2 
 
 IS 
 
 
 3. Find the product of 
 
 a a a a 
 
 and 
 
 ahhh 
 
 
 a b c c 
 
 
 abed 
 
 
 -110 
 0-1 1 
 0-1 1 
 
 111-1 
 
 , and 
 
 thence show that the first determinant = a (b—a) (c—b)(d—c), 
 4. Show that 
 
 1111 
 -1-1 1 1 
 -1 1-1 1 
 -1 1 1-1 
 
 a 
 
 h 
 
 c 
 
 d 
 
 X 
 
 h 
 
 —a 
 
 d 
 
 c 
 
 
 c 
 
 d 
 
 —a 
 
 b 
 
 
 d 
 
 c 
 
 b 
 
 —a 
 
 
 b-\-c-i-d—a 
 b—a-\-d-\-c 
 c+d—a-\-b 
 d-^c-{-b—a 
 
 a—b-{-c+d 
 ■ b-^a-{-d-\-c 
 ■c—d—a+b 
 d—c+b—a 
 
 a-\-b—c+d 
 —b—a—d-\-c 
 —c-\-d-\-a+b 
 — d-\-c—b—a 
 
 a+b-\-c—d 
 — b — a-\-d—c 
 —c-}-d—a—b 
 —d-\-c-\-b-\-a 
 
54 
 
 THEORY OF DETERMINANTS. 
 
 = (6+c+cZ-a) (c+d+a—b) (d-j-a+b-c) {a-\-b-\-c—d) 
 1111 
 
 1 
 
 1 
 
 -1 
 
 -1 
 
 1 
 
 -1 
 
 1 
 
 -1 
 
 1 
 
 -1 
 
 -1 
 
 1 
 
 and thence show that the first determinant 
 = (b-\-c-\-d—a) (c-i-d+a-b) (d+a+b—c) (a+b+c—d), 
 5. Show that 
 
 1 a^+a^ -2a -^2a 
 
 1 b^-\-l3^ -26 -2(3 
 
 1 c2+y2 _2c -2y 
 
 1 d2+82 -2d -28 
 
 a^+a^ 1 a a 
 b^+fS' 1 b /3 
 c^+Z 1 c y 
 ^2+ 8M d a 
 
 
 
 {a-cy-{-{a-yr 
 la-dy-\-(a-8y 
 
 (a 
 
 
 
 (b-cy+(i3-yy 
 
 
 
 (c-dy+(y-sy 
 
 (a-dy+(a-&y 
 (b-dy+(^^By 
 (c-dy+{y-By 
 
 
 
 6. Show that 
 
 
 
 
 ai 5i Ci 1 
 
 X 
 
 1 fci 
 
 X 
 
 10 
 
 ttg &2 ^2 1 
 
 
 1 A;2 
 
 
 10 
 
 as 63 C3 1 
 
 
 1 A:3 
 
 
 10 
 
 1110 
 
 
 1 
 
 
 hi Jiz /is 1 
 
 equals the determinant in Example 12, page 38. 
 7. Find the two determinant factors of 
 
 «i Mi+Ci2/i ^2+^12/2 ; also of 
 
 «2 b<^i'-{-c^i 62^2+^22/2 
 
 Cts Ml + Cs^l &3aJ2+C3y2 
 
 8. Form the product of 
 
 % 61 
 
 Og 62 
 
 axi-\-cZi fxi+gzi 
 
 ax2+by2-\-cz2 dy^ fx2+gz2 
 
 bys+czs dys gz^ 
 
 and 
 
 ai A ri 
 
 "2 A 72 
 as /?8 78 
 
 The order of the first determinant may be raised to that of 
 
 the second by writing it 
 
 0^2 ^2 ^2 
 
 1 
 
 and the product can then 
 
GENERAL PROPERTIES OF DETERMINANTS. 
 
 55 
 
 be found in the usual way. If we wish the product to contain 
 only the elements found in the two factors, how should the first 
 determinant be written? 
 
 From this example it is evident that the product of any num- 
 ber of determinants of different degrees can he expressed as a 
 determinant of the nth degree.^ n being the highest degree among 
 the factors. 
 
 9. Emplc 
 
 \ — c-\-id 
 may be written 
 
 oying the notation i = V— 1, shaV \bsX tiie product of 
 a-\-ih c-j-id and [ ai—ihi Cj — idJ 
 
 a — ib 
 
 I D-iC 
 -B-iA 
 
 \-Cy 
 
 B^iA 
 D + iC 
 
 C = abi — aib + cdi — Cid 
 D = aa-i + bbi + cc^ -}- ddi ; 
 
 m which 
 
 A = bc-i — biC + adi — aid 
 
 B = cai — c^a + bd^ — bid 
 and thence show that the product of two sums, each of four 
 squares, is itself the sum of four squares, (Euler's Theorem.) 
 
 10. Show (1) that the product of {aiboCs] and \piq2Ts\ may 
 be expressed as a determinant of the fourth order by writing 
 the two factors cij bi Cj and — i^i ^i rj respectively. 
 
 cii bi Ci 
 
 and — 
 
 a2 62 C2 
 «3 h C3 
 1 
 
 
 Pi Qi 
 
 
 
 n 
 
 P2 q2 
 
 
 
 ^2 
 
 Ps Qs 
 
 
 
 ^3 
 
 
 
 1 
 
 
 
 (2) By writing the two factors 
 
 Pi qi n 
 P2 O q2 rg 
 Ps qs Vs 
 10 
 10 
 show that the product is a determinant of the fifth order. 
 
 (3) By writing the two factors 
 
 «! bi Ci 
 
 and 
 
 ^2 62 C2 
 
 
 as bs Cs 
 
 
 10 
 
 
 1 
 
 
 tti 61 Ci 
 
 and 
 
 aa 62 C2 
 
 
 asbsCsO 
 
 
 10 
 
 
 10 
 
 
 1 
 
 
 10 
 
 10 
 
 10 
 
 piqiri 
 
 P2 qz r2 
 
 Ps qs n 
 
* SI 
 
 66 THEORY OF DETERMINANTS. 
 
 show that the product is a determinant of the sixth order. 
 This example, and the theorems of 51 and 53, show that the 
 product of tivo determinants of the nth order can be eoopressed as 
 a determinant of each of the following orders: nth, (n + l)^A, 
 (w+2)«7i... (2w-l)«7i, 2nth, 
 
 55. Theorem. — Any determinant A may be expanded as a 
 sum of products of pairs of minors. The first factor of each 
 product is a minor of the rth degree, formed from a set of r 
 chosen rows, and the other factor is the complementary minor of 
 the first factor . The sign of a product is -f- or — , according as 
 the product of the ^jrincipal terms of the factors regarded as a 
 term of A is -j- or — .* 
 
 Every term of A contains r elements from the columns of a 
 set of r columns found in the n columns and first r rows of A. 
 That is to say, from every minor of the rth degree formed from 
 the first r rows, r I partial terms of A can be formed. Now, 
 the remaining (n—r) elements of every such partial term will 
 be found in the remaining rows and columns after removing 
 one of these minors of the rth degree. Or, in other words, 
 (n— ?') ! partial terms of A corresponding to the r ! other par- 
 tial terms are found in every minor complementary to one of 
 
 the first set. — ; — minors of the rth degree can be formed 
 
 r ! (n— r) ! 
 
 from the first r rows. Now, the product of two such comple- 
 mentary minors gives r ! (n— r) I terms of A ; consequently, the 
 sum of all the products gives n ! terms, i.e., the full number of 
 terms in A.- 
 
 To fix the sign of any product in this expansion, we have 
 only to remember that its sign must be the same as the sign of 
 the product of the principal terms of the two minors. This 
 latter product being a term of A, the sign of the product of the 
 two minors must be the sign of the product of their principal 
 terms, regarded as a term of A. 
 
 If the selected rows are not the first r rows, we can easily 
 make them so ; then, after giving A the proper sign factor, the 
 demonstration applies as given. 
 
 * Tills expansion is knowr *^ Tiaplace's Theorem. 
 
GENERAL PROPERTIES OF DETERMINANTS. 57 
 
 Illustrations : 
 
 Selecting the first two rows in ki h^ c^d^]-, we have 
 
 4- 1 6i Cgl ^3^4! — I &i cZgl I tto C4I 4- 1 Ci dgl I CI3&4I . 
 Let the student select the first two columns of 1% 62 C3 (^4! , and 
 expand, obtaining 
 
 \(hh\\c3di\ — \aM\c2di\-{-\aM\C2ds\-h\a2bs\\cM 
 — lazb^WcM-hldsbiWcidzl. 
 Show that 
 loihcsd^esl = — lazb^l \cidse5\-{-\a.jC^\ Ibidse^l — laid^l IbiC^e^l 
 + \a2ei\ 16103^51 — 16204! laid^e^l + lbzd^l la^c^e^l 
 — 16264! !aiC3d;5! + 10264! Iai63d5l4-!c2e4l laib^dgi 
 -Id^e.l \a,bsc,\. > !^^/^y/^SvyS' 
 
 What is the relation of 41 to the present theorem? . -' * 
 
 56. It will be interesting to note what results, if, instead of 
 multiplying the minors of the rth degree formed from r chosen 
 lines by their complementaries, as in the last article, we mul- 
 tiply every such minor by the complementary of a corresponding 
 minor formed from r lines different from those first chosen. 
 By the preceding article 
 jai 6203^465! 
 
 = !ai62l !c3d4e5! — !ai63l \c2d^e^\ + !ai64! \c2dse5\ — \aM \c2d.^ei\ 
 
 + ka 63! !cid4e5! — |a264! 101^365! + |a265l 101^364! -I- |0364l|cirZ2e5l 
 
 — Ia365l !cid!2e4! + !«46^ IcidzS^l. 
 
 Now, if in the above we write c for 6, it is evident that the 
 determinant on the left vanishes, and hence the second mem- 
 ber vanishes ; but by this substitution we multiply the minors 
 formed from the ^?'s^ and third columns of I a^ 62 03^/465 1 by the 
 complementaries of the corresponding minors formed from the 
 Jirst and second columns. It is obvious that the truth here 
 exemplified holds in general. Moreover, it includes the special 
 case of 45. 
 
68 
 
 THEORY OF DETERMINANTS. 
 
 In symbols, the expansion of a determinant by 55 is expressed 
 
 by writing 
 
 A = :S|ap6J Aflp,6^, 
 
 where the chosen columns are two in number, or 
 
 A=:Slap&gCj Aaj„6j,c„ 
 
 where the chosen columns are three in number ; and so on. 
 
 Employing the notation of double subscripts, we have, in 
 general, 
 
 57. Theorem. — The product of a determinant A = I ai„| , and 
 any one of its minors M, of order m, is a determinant A' of 
 order n -\- m. A' is expressible as the sum of pi'oducts of pairs 
 of minors of ^; the first factor of each product is a minor of A, 
 formed from r chosen rows containing J/, and the second factor 
 is that minor of A containing the complementary of the first 
 factor and the minor M. The sign of each product is determined 
 as in 55. 
 
 Let the chosen rows referred to in the statement of the 
 theorem be the first r ; then, by 51, we have at once 
 A' = 
 
 «11 %2 .-.^l*-! 
 
 «1A .-.air-l 
 
 Ctlr 
 
 «lr+l • 
 
 ••«!« 
 
 .. 
 
 . 
 
 
 
 ttai %2 "-ailc-l 
 
 «2Jt ...agr-l 
 
 a2r 
 
 «2.+l • 
 
 ..a,, .. 
 
 . 
 
 
 
 
 
 
 
 ... 
 
 «Ar+l • 
 
 ..a,„ .. 
 
 . 
 
 
 
 ^Al <^A2 '"^kk-\ 
 
 a,ac •••«*r-l 
 
 a*r 
 
 <^k+l\(^k+\J'"(^k-h\k-l 
 
 (^k+yc • '(^k+i/-- 
 
 -1 ^ft+lr 
 
 ^A+Ir+1* 
 
 ..a.+i. .. 
 
 . 
 
 
 
 
 
 
 
 ... 
 
 ... 
 
 
 
 . ... 
 
 ... 
 
 ^r-ll^r-12'"^r-lA:-l 
 
 Ov_^...a,_i,. 
 
 -ittr-lr 
 
 «y-ir+i. 
 
 ..«.-!„ .. 
 
 . 
 
 
 
 a,l a^ "'Ctrk~l 
 
 aru ...a.r-1 
 
 a„ 
 
 a^r+i ' 
 
 
 . 
 
 'Ctr+lr- 
 
 
 
 1 ^r+lr 
 
 ^r+uf^r+l2" '(^r+lk~l 
 
 Ctr+lk-"<^r+lr- 
 
 -l«r+lr 
 
 » 
 
 
 
 ... 
 
 ... 
 
 
 
 . ... 
 
 ... 
 
 «nl an2 —Clnk-l 
 
 ... 
 
 Ctnr 
 
 
 ttnr+l • 
 . 
 
 -a^n 
 
 Clnk '. 
 
 -a^r-i 
 
 «nr 
 
 ... 
 
 .. 
 
 a^ .. 
 
 'C(kr-1 
 
 a*r 
 
 ... 
 
 ... 
 
 
 
 . 
 
 .. 
 
 Ctk+lk" 
 
 'dk+lr. 
 
 •iCti+lr 
 
 
 
 
 
 ... 
 
 ... 
 
 
 
 
 
 . ... 
 
 ... 
 
 ... 
 
 ... 
 
 
 
 . 
 
 .. 
 
 Clr-lk" 
 
 •«r-lr- 
 
 -iCt'r-lr 
 
 ... 
 
 ... 
 
 
 
 . 
 
 .. 
 
 ark •• 
 
 •«rr-l 
 
 «rr 
 
GENERAL PROPERTIES OF DETERMINANTS. 
 
 59 
 
 where the minor by which A is multiplied is enclosed. Further, 
 observe that the n — r rows of A not included in the chosen 
 rows are prolonged in A' with the elements of these same rows 
 repeated in order of the columns beginning with the kth. Now 
 add the A;th row of A' to the {n + l)th, the (fc -f- l)th row to 
 the {n + 2)th, and so on, finally adding the rth row to the last. 
 Afterward subtract the (n + l)th column of A' from the A:th, 
 the (w + 2)th column from the (A;+l)th, and so on, finally 
 subtracting the last column from the rth. Then 
 
 «11 
 
 012 
 
 ...ttli-l 
 
 f'l* 
 
 ...ai^_i Oi^ 
 
 %r+l 
 
 ...«u 
 
 «21 
 
 022 
 
 •••«2*-l 
 
 «2* 
 
 ...a.2r-l Oj2r 
 
 «2r+l 
 
 ...asn 
 
 %1 ^*2 "'^kk-l 
 ^r+1 1 ^r+1 2 • • • ^r+1 A-1 ^ 
 
 <^r-ll<^r-12"'<^r-l&-l ^ 
 
 «'U .• 
 
 .a^r-i 
 
 «■*. 
 
 a^+u- 
 
 •«mr- 
 
 -1 (^k+\r 
 
 
 
 . ... 
 
 ... 
 
 «,_!,.. 
 
 .a.^ir- 
 
 -l«,-lr 
 
 a,, .. 
 
 .a,,_, 
 
 a,. 
 
 f'ftr+l 
 
 ''A:+ln 
 
 ^r-lr+l'"^r-l» ^ 
 
 ■r+1 
 
 
 
 
 
 
 
 
 
 
 ... 
 
 ... 
 
 
 
 
 
 
 
 
 
 
 ... 
 
 
 
 
 
 ... 
 
 
 
 
 
 ... 
 
 
 
 
 
 ... 
 
 
 
 
 
 
 
 ... 
 
 ... 
 
 ... 
 
 
 
 
 
 ... 
 
 
 
 
 
 a.+ii...o. 
 
 +lr- 
 
 l«r+l,- 
 
 
 
 ... 
 
 ... 
 
 «nt •••«« 
 
 r-V 
 
 (^nr 
 
 (^kk •••«* 
 
 r-1 
 
 Ctkr 
 
 <^t+lA*"<^A+lr- 
 
 l«A+lr 
 
 
 
 ... 
 
 ... 
 
 ar-W'Clr 
 
 -Ir- 
 
 l«r-lr 
 
 a,j, ...a. 
 
 r-1 
 
 a,. 
 
 By 55 A' can be decomposed into products of pairs of minors, 
 viz., the minors of the rth order formed from the first r rows 
 and their complementaries. Since the elements in the columns 
 of A' directly below M are zeros, all the minors of the rth 
 order,. formed from the first r rows, will have complementaries 
 that vanish unless the said minors contain the given minor M. 
 Hence the first factors of the products in the expansion of A' 
 will all be minors of A, of the rth order, that contain the given 
 
60 
 
 THEORY OF DETERMINANTS. 
 
 «1 
 
 &i 
 
 Ci di 
 
 6,0 
 
 
 
 ^2 
 
 &o 
 
 C2 (^2 
 
 ^2 
 
 
 
 a-i 
 
 bs 
 
 Cg <^3 
 
 63 
 
 
 
 a. 
 
 h. 
 
 C4 d^ 
 
 64 64 
 
 C4 
 
 as 
 
 h 
 
 C5 d. 
 
 65 h 
 
 C5 
 
 
 
 
 
 
 
 62 
 
 C2 
 
 
 
 
 
 
 
 63 
 
 C3 
 
 minor. Further, each complementary of such a factor is made 
 up of the n — r rows of A not found in the first factor and the 
 r — Tc + l rows in which M is found. Which proves the 
 theorem. 
 
 I«1&2C3C24 65! 1^203! = 
 
 tta b^ C3 0^3 63 U 
 
 a^ 64 C4 (^4 64 64 C4 
 
 1^162^31 \d^e^h2C^\ 
 16102^3! I a4 656203! 
 16102631104^562031 ; 
 
 !«! 62 63(^4 65! 16203(^4! = 10162630541 1656203^4! +161020^3641 10562 03^41. 
 
 The student may show (change the rows into columns before 
 applying the theorem) 
 
 1016263^465! I63C4I = 1016263^4! 1636465! — 1016364(^5! I636462I 
 
 + 1026304(^5116364611; 
 
 1016263^4651(^2 
 = 161(^263! la4&6C?2l — \cid2e^\ \a^hsd^\ + 161(^265! 10364(^21 
 + l62C?364l loi65c22l " \<^2di^5\ lo, 64(^21 + Uid^e^l \aibMi 
 
 = — 1^162! 1036465(^2! + 1(^263! !«! 6465(^2! — 1(^264! IOi6365C?2l 
 
 + lc?2e5ll«i 6364(^2! » 
 
 The second illustration given is especially interesting as it 
 shows the form of the product when the minor is of order n — 2. 
 In that case the chosen rows are 7i — 1 in number, and the 
 development consists only of two terms, each term being the 
 product of two determinants of the (71— l)th order. If we 
 change the order of rows and columns in the result, we have 
 
 !«! 6263(^4 65! 16263(^4! = 10162 63(^4! 16263(^465! — 16162(^36411026364(^5!, 
 
 or A Aai,e6 = Ae5 Aa^ — Aa^ Ag, ; 
 
 and, in general, 
 
 A Aa,7fc, apq = Aart ^Opq — Aa,-j Aop*. 
 
 Employing an obvious extension of the notation described in 
 the latter part of 39, the last formula becomes 
 
 A ^^^ = i?^ ^ _ ^ -^ 
 dOi 
 
 dcii^ da da^ da 
 
 -iq ^%k 
 
GENERAL PROPERTIES OF DETERMINANTS. 61 
 
 Rectangular Arrays or Matrices. 
 
 68. As a determinant is a function of n^ quantities, the 
 elements are always found in a square array. It is often 
 necessary to consider the determinant obtained by applying the 
 process of 53 to two rectangular arrays of elements, i.e., arrays 
 in which the number of rows is not equal to the number of 
 columns. We will now investigate the value of this product. 
 
 1st. When the number of columns exceeds the number of 
 rows: 
 
 The product of two arrays (matrices) of elements in whicli the 
 number of columns (m) exceeds the number of rows (n), is a 
 determinant which is equal to the sum of all the products in which 
 the first factor is a determinant of the nth order formed from the 
 first array (matrix) , and the second factor is the corresponding 
 determinant of the nth order formed from the other array 
 (matrix) . Let the two arrays of elements be 
 
 «ii «i2 ••• <^in ••• ^im 1 an ai2 ••• ^in ••• aim 1 
 
 Cl2i a^ ... a2n . . . a2m I g^jj J a21 0-22 • • • a.2n • • • CL2m I n <1 m 
 
 <^nl «'m2 ••• Ct„„ ... «„„, J a„i a„2 •.. Clnn ••• CLnm J 
 
 Applying the process of 53, we have the determinant 
 
 ^nau4-*'*+<^'l»ainH h^lw "im <^llf^21-\-"--\-CLln<hn 
 
 <^21 an H h a2«ai„ H f- «2m ai^ ^21 a2i H h Of 2na2n 
 
 <^niaiiH httn^a^.H ha„„ai„ ania2iH ha^^og^ 
 
 H \-(^imf^2m •'• «na„iH |-ai„a„„H \-ai^a„„ 
 
 -h"'-^-Cl2ni"-2m "' «2ia„iH \- Ci2n O-nn -\ 1-«2„ a„„ 
 
 H \-(^nm<^2m '" «„ia„iH \-Ctnn"-nn-\ \- <^'nm O-n,, 
 
 Now we may form from A a number of determinants A,, Ag, A3 ••• 
 of the nth order, the elements of which are all polynomials con- 
 
62 THEORY OF DETER^UKANTS. 
 
 sistiug of n terms each. The number of such determinants is, 
 
 of course, m(m - 1) (m-2) - (m-« + 1) . ^et us consider 
 
 n! 
 one of these determinants ; take, for example Ai, whose columns 
 are formed from the first n terms in the columns of A. AVe 
 have, accordingly, 
 
 Ai= auan4-C^i2ai2-| h^inain 
 
 «2ian + <^22ai2H |-«2nain 
 
 ^n\ "11 + 0tn2 ^12 H f" ^nvP-\n 
 
 <^lia21 + <^12«22H \-(^\nf^2n '" Ctji «,»! + « 12 an2H h^lnOnn 
 
 «21 «21 + ^22«22+*"+^2na2» •*" ^<21 a„i + a22 a„2 H |-«2»ann 
 
 «nia21 + <^n2a22H \-<^nnO-27i '" «»ianl + ««2aH2H h^nnttnn 
 
 Now Ai is, by 53, the product of two factors, the first of which 
 is the determinant formed from the first n columns of the first 
 array of elements, and the second is the determinant formed 
 from the corresponding n columns of the second array. In a 
 similar manner we may show, that each of the determinants 
 Ai, Aa, Ag--- is the product of two factors, each factor being a 
 determinant formed from n corresponding columns of the two 
 given arrays. Then in order to establish the proposition it 
 
 remains to be shown that A = Ai + Ag + Ag H . Each of the 
 
 determinants Ai, Ag, Ag--- can be decomposed into n! non- 
 vanishing determinants whose elements are monomials. Ac- 
 cordingly the sum Aj -j- As + AgH — will contain 
 m(m— l)(m — 2) ••- {m — n-{- 1) 
 non-vanishing determinants whose elements are monomials. 
 Returning to A, we see that it can obviously be decomposed 
 into m" monomial element determinants ; but those which do 
 not vanish are only m(m — l) (m— 2)--- (m — n-f 1) in number. 
 Now observing that each one of these monomial element deter- 
 minants is a jmrt of that one in the series Aj, A2, Ag--- in which 
 its columns occur as parts of columns, the proposition is estab- 
 lished. 
 
GENERAL PROPERTIES OF DETERMINANTS. 63 
 
 Illustration : 
 
 Performing the operation of 53 upon 
 
 ag 02 Co 0,2 j a2 P2 72 f>2 j 
 
 we obtain the determinant 
 
 «2ai+ ^2A + C2yiH-(^2Sl «2a2 4-&2ft +^^272 + ^2^2 
 
 This determinant the student can readil}^ show is equal to 
 
 («i W («1 A) + (^^2) (aiy2) + («id2) (ai82) 
 + (2^1 C2) (A 72) + (&lC^2) (/?lS2) + {cA) (7182) . 
 2d. Wlie7i the number of rows exceeds the number of columns. 
 Consider the two arrays. 
 
 a^ &2 f and ttg ^2 
 
 Multiplying as before, we have 
 
 ajai + by^^ aia2 + &1/32 ajag + ftjft = 0. 
 
 «2ai + i^2/?l «2a2 + &2/?2 <^2 ^3 + h(^S 
 Cts"! + ^sA «3a2 + ^sft WsCtg 4- 63^3 
 
 The value of A is readily seen to be zero when we notice that 
 it can be obtained by multiplying two determinants formed from 
 the two given arrays by prefixing a column of zeros to each. The 
 method of proof employed in this special case is general. It is 
 only necessary to add to each array as many columns of zeros as 
 are necessary to make each array square, and then compare the 
 product of the two determinants thus formed with the deter- 
 minant formed by compounding the two matrices. 
 
 Reciprocal Determinants.* 
 
 59. If the principal minors of the elements of a determinant 
 are themselves made the corresponding elements of another 
 
 * Reciprocal determinants would more properly be considered in the 
 next chapter since they are among the " special forms," but for several 
 reasons it is thought best to introduce them here. 
 
64 
 
 THEORY OF DETERMINANTS. 
 
 determinant, the determinant thus formed is called the reciprocal 
 or adjugate determinant. Or, in other words, the elements of 
 the reciprocal determinant are the complementary minors of the 
 corresponding elements in the original determinant. 
 
 The reciprocal of {cti h^ c^) is 
 
 ipp 
 
 
 
 (as Cg) (tto h) 
 
 KC3) -(ai^a) 
 (ai C2) (ai 62) 
 
 Assimilating the notation of 19, we have 
 
 \AiB2Cs,..L,\, 1^1,1, or 1^11 ^22 -433... A„l, 
 
 for the determinant adjugate to 
 
 I «i h C3 ... ^n I? I «i^ |, or 1 an a22 %3 •.• <^nu|? 
 respectively. 
 
 If the minus signs in the first illustration are erased, what is 
 the effect upon the determinant? How is it in general? 
 
 60. Theorem. — The determinant A' adjugate to any deter- 
 minant A oftJie nth degree, equals the (n — l)th power of A. 
 
 We have, for example. 
 
 Whence 
 
 A = 
 
 «! 61 Ci 
 
 
 ^2 ^2 C2 
 
 
 «3 h C3 
 
 AA' = 
 
 A = 
 
 
 A 
 
 
 A 
 
 and A'= 
 
 A^ 
 
 A B, C\ 
 
 A2 B, (72 
 
 Ao Bo C.Q 
 
 The process here exemplified is perfectly general, hence the 
 proposition. 
 
 61. Theorem. — Any minor of the kth degree of the reciprocal 
 determinant A' is equal to the complementary of the corresponding 
 minor in the original determinant A multiplied by the {k — \)th 
 power of A. 
 
 Let A = I ai4 1, and A'= I ^14 1 . 
 
GENERAL PBOPERTIES OF DETERMINANTS. 
 
 65 
 
 Transform A and A' so that the minors | ctn ^32 «44 ] and 
 I All -^32 ^44 I occupy the first three rows and columns in their 
 respective determinants. Then 
 
 A=( — 1)'' ail «i2 
 
 ^31 ^^32 ^ai 
 41 42 
 
 and A'=(-l)'^ 
 
 An 
 
 Ai, 
 
 ^4 
 
 ^1, 
 
 
 A. 
 
 As, 
 
 ^4 34 
 
 ^33 
 
 
 -I41 
 
 ^442 
 
 -i« 
 
 ^« 
 
 
 An 
 
 A.^_ 
 
 A.^ 
 
 ^.. 
 
 Then 
 
 
 ■ 
 
 
 lAl A2 ^44|=(-l)'^ 
 
 Ai 
 
 Ay2 
 
 A,, 
 
 A„ 
 
 
 Ai 
 
 A,, 
 
 Au 
 
 •^33 
 
 
 A,i 
 
 Aj^ 
 
 Au 
 
 ^43 
 
 
 
 
 
 
 
 
 1 
 
 Multiplying, 
 
 
 
 
 A\Aii A^ A^\=z 
 
 A 
 
 
 
 «13 
 
 = a2C 
 
 
 
 
 A 
 
 «83 
 
 
 
 
 
 A 
 
 "c 
 
 
 
 
 
 
 
 a^ 
 
 
 Whence | An ^433^44 ] = aog A^, which is the required value of a 
 first minor of A'. 
 
 To find the value of a second minor of A' we may proceed as 
 follows : 
 
 The minor 
 
 1-^22 As, 
 
 (-l)nA2 
 
 •^32 
 
 
 
 
 -"■2; 
 
 A 
 
 
 
 ■^21 
 
 A^i 
 1 
 
 
 A4 
 
 -^34 
 
 
 
 1 
 
 and the corresponding form of A is 
 
 (-1) 
 
 «'23 
 
 a^i 
 
 a24 
 
 «33 
 
 asi 
 
 (X34 
 
 ^'13 
 
 «n 
 
 «14 
 
 «43 
 
 «41 
 
 «44 
 
QQ THEOBY OF DETEEMINANTS. 
 
 As before, 
 
 A Moo AJ = 
 
 A^ ail a. 
 
 A 0821 «2- , _ 
 
 A agi a34 
 an an 
 a^i a^ 
 
 Whence, |^22 ^33!= \<^n «44|^- 
 
 The student may put 
 
 A = I ai 62 C3 di I and A' =\Ai B^ C^ A | » 
 
 and then show that 
 
 \B, a, A|=aiA2; 
 
 Ml A|= I&2 C3IA. 
 
 The general theorem, of which the preceding are special cases, 
 is proved as follows : 
 
 Let A =|ai^| and A'= |An|? 
 
 and let the minor of the Mh order of A' whose value is sousfht be 
 
 V= 
 
 ^PiQi ^PiQz ^PiQs •' 
 
 ^PiQk 
 
 ^PiQl ^P2l2 ^P2Q3 '" ^P2<lk 
 
 \^PkQi ^Pk<l2 ^PkQa '" ^PkQk^ 
 
 Now putting 
 
 fi =Pi +P2 + ••• +Pu + qi + q2-\ f- ^*, 
 
 we may write 
 
 Qp^q^ ... Op^g^ ap^i ... Ctjy^q^-l Ctpiq^+l ••• <Vi?2-l ^Pi?2+1 " 
 ^PiQi '" ^PiQk ^Pi'i- '" ^P2Qi-l ^zQ-i+l ••• ^i>2(?2-l ^i'2Q'2+l •' 
 
 ^PkQi ••• ^PkQk ^Pk^'"' ^i'ftffi-l ^PkQi+'^ '" ^Pk^2-^ ^*ff2+l •' 
 
 Cip^n 
 
 ^Pk^ 
 
 aiq^ ... aig-^ an ... a\q^-\ aig-^+i ... aig^-i ai^^+i ... ai» 
 
 a2g^ ... a2g^ ttsi ... a2g'j-l O^g^i+l ••• ^2^2-1 ^ffa+l ••• «2n 
 
 * In this determinant the subscripts }\, }\, p^, ... g^, q^, q^, ... of course 
 stand for any integers in order of magnitude. 
 
GENEEAL PROPERTIES OF DETERMINANTS. 67 
 
 The coiTesponding form of A;^ is 
 
 . . . Ap^q^i Ap^q^+i . . . Ap^n 
 
 • • • ■4^2?2-l Ap^q^+l . . . ^^271 
 
 • • • Ap^q^-1 Ap^q^+i . . . Ap^n 
 ... ... 
 
 ... ... 
 
 ApiQi ' 
 
 • • Ap^qi^ 
 
 Axl" 
 
 • Ap^q^- 
 
 -lAp,q,+l 
 
 Ap^Qi ' 
 
 ••Ap^q„ 
 
 Ap^\'' 
 
 ' Ap^q^~ 
 
 -1 Ap^q^+i 
 
 Apj,gj_ • 
 
 ' • Ap^q^ 
 
 Aa-1" 
 
 Apf^Qi- 
 
 -i4pftgi+i 
 
 . 
 
 .. 
 
 1 .. 
 
 
 
 
 
 . 
 
 . 
 
 .. 
 
 
 
 
 
 . 
 
 . 
 
 .. 
 
 1 
 
 
 
 . 
 
 . 
 
 .. 
 
 
 
 1 
 
 . 
 
 . 
 
 .. 
 
 
 
 
 
 . 
 
 . 
 
 .. 
 
 
 
 
 
 . 
 
 . 
 
 .. 
 
 
 
 
 
 
 
 . 
 
 .. 
 
 
 
 . 
 
 .. 
 
 1 
 
 . 
 
 .. 
 
 
 
 1 . 
 
 .. 
 
 
 
 •••l(n-fc) 
 
 We notice that this form of A;^ is just the same as if it had 
 been derived from A' b}^ making the pith, pgth, •••^^th rows of 
 A' the 1st, 2d, ••• A:th rows, making the same changes in the places 
 of the ^ith, goth, ... Qj^th columns, and then putting 1 for each 
 remaining element of the principal diagonal, and for every 
 other element of the n — k rows of which A;^ is not a part. 
 
 Multiplying, we have 
 
 AA, 
 
 = A . 
 A . 
 
 ttp^i ... Clp^q^-l Ctpi^i+l ••• (^p^q^-l ^PiQi+l '" ^Pi^ 
 
 
 
 
 • 
 
 
 
 
 
 ... A ap^i.. 
 
 • ^PkQi-T^ "2^i-?i+l • 
 
 •• ^hk<li-'^ ^^PkQi+'i- 
 
 
 
 ... an . 
 
 • «lgi-l «l(Zi+l • 
 
 •. «1(72-1 Cllq^+i 
 
 
 
 ... a2i .. 
 
 • «2f?i-l «2(7i+l • 
 
 " «2(72-l Ct2g,+1 
 
 ... a„i ... Clngi-l CLnq^+l ••• ^n^j-l ^^nq^+l 
 
 ••• Cipkn 
 
 ... ain 
 
 ... a-zn 
 
 • . . Cf/fin 
 
68 THEORY OF DETERMINANTS. 
 
 Now this determinant is at once expressible as the product 
 of two determinate factors, and we have 
 
 A\ — A* times the complementary of the tninor of A corre- 
 sponding to A^ in A'. 
 Whence 
 
 A;^ = A*"-^ times the complementary of the minor of A corre- 
 sponding to A„of A', 
 as was to be shown. 
 
 62. From the preceding article it follows at once that if 
 
 A = 0, then 
 
 ••=0; 
 
 
 An Au 
 
 -^21 -^22 
 
 = 
 
 
 = 
 
 An Aio 
 
 Aqi Aq2 
 
 i.e., in general 
 
 whence 
 
 Aij, Ape 
 
 
 0, 
 
 
 
 or A, J, : Aie ::Ap^: Ap^. 
 
 That is to say : 
 
 If A = 0, the cof actors of the elements of any row are propor- 
 tional to the cofactors of the corresponding elements of any other 
 row. 
 
 From the preceding article we have also 
 
 = A X complementary minor of 
 
 Aik Ai. 
 
 ■^^pk -^^pe 
 
 which may be written 
 
 
 
 dA dA 
 dajj, da/g 
 
 dai^dap^ ' 
 
 
 
 dA dA 
 
 
 
 
 dap, dap. 
 
 
 
 whence 
 
 da 
 
 1'A dA dA 
 ,,dape da,, da^. 
 
 dA dA 
 dap, da^. 
 
 which is the formula already obtained in 57. 
 
GENERAL PROPERTIES OF DETERMINANTS. 
 
 69 
 
 1. Show that 
 0, 
 
 EXAMPLES. 
 
 also 
 
 2. Show that 
 
 
 
 
 ^2 
 ^1 
 
 
 
 
 
 
 
 
 h 
 
 h 
 h 
 
 \B, 
 
 
 
 
 2/3 
 2/2 
 
 2/1 
 
 
 a4 
 
 
 
 
 
 
 
 
 = -2!i|a;i2/2%l |2/i2^2|; 
 
 |«1&4| |0t2&5l |«3&6 
 
 C2 
 
 
 
 
 
 ^3 
 
 
 
 
 
 ClA 
 
 A 
 
 A 
 
 CiB, 
 
 B, 
 
 B, 
 
 c,C\ 
 
 c. 
 
 c. 
 
 = \A^B^C^\ lai^gCs]. 
 
 3. If A is a determinant of the nth order, having n — m zero 
 elements in the corresponding places of m rows, then A is the 
 product of that minor whose elements are the other elements of 
 the 771 rows and its complementary ; the sign of the product is 
 determined as in 55. 
 
 4. If any determinant of the nth order has more than {n — m) 
 zero elements in the corresponding places of m rows, the deter- 
 minant vanishes. 
 
 5. 
 
 «! 
 
 \ 
 
 Cy 
 
 M 
 
 iV^ 
 
 = 
 
 a^-M 
 
 h^-N 
 
 0^2 
 
 h 
 
 C2 
 
 P Q 
 
 
 a,-P 
 
 h-Q 
 
 0^3 
 
 h 
 
 C3 
 
 ttg 63 
 
 
 
 a^ 
 
 h 
 
 ^'4 
 
 a4 64 
 
 
 
 as 
 
 h 
 
 C5 
 
 as 65 
 
 
 
 «! 
 
 a^ 
 
 ag 
 
 a. 
 
 = \d,a,\ \h,c\ 
 
 
 &i 
 
 
 
 
 
 h 
 
 
 
 Ci 
 
 
 
 
 
 C4 
 
 
 
 d. 
 
 d. 
 
 d. 
 
 C^4 
 
 
 
 
 
 03^4651; 
 
70 
 
 THEORY OF DETERMINANTS. 
 
 6. 
 
 tti 
 
 «2 
 
 a. 
 
 ^4 
 
 «5 
 
 «6 
 
 Oj as 
 
 ag 
 
 «10 
 
 
 h 
 
 h 
 
 h 
 
 &4 
 
 ?>-, 
 
 h 
 
 h h 
 
 &9 
 
 K 
 
 
 
 
 
 
 
 
 
 
 
 
 Ce 
 
 Cr Cg 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 f?6 
 
 cZ; dg 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ec 
 
 e? eg 
 
 
 
 
 
 
 /l 
 
 /. 
 
 f. 
 
 /. 
 
 /. 
 
 ./o 
 
 /r /s 
 
 
 
 
 
 
 9i 
 
 92 
 
 9s 
 
 i/4 
 
 i/o 
 
 ^6 
 
 i/z ^8 
 
 
 
 
 
 
 \ 
 
 h 
 
 h 
 
 h 
 
 h 
 
 /'« 
 
 /'r /'8 
 
 
 
 
 
 
 h 
 
 h . 
 
 
 
 
 
 
 
 ?'(; 
 
 ^T ^*8 
 
 
 
 
 
 
 k. 
 
 A:2 
 
 
 
 
 
 
 
 A'6 
 
 Av A-g 
 
 
 
 
 
 = 
 
 = -| 
 
 <-hK\ 1^6 
 
 CZyCg 
 
 IIA 
 
 74/^.1 \hK 
 
 X 
 
 
 7. Show 
 
 tlint 
 
 
 
 
 
 
 
 
 
 {ay-h^y (a.-boY 
 
 (a,-b,r {a,-b,r 
 {a,-b,r {a,-b.^' 
 
 {a,-b,y 
 {a,-b,y 
 {a^-bsY 
 
 K- 
 
 -b.y 
 
 («.- 
 
 -hy 
 
 («3- 
 
 -b.y 
 
 (a,- 
 
 -hy 
 
 = 0. 
 
 This may be proved by multiplying the two arrays : 
 
 and 
 
 8. Show that 
 
 |«ln| (^1 + 3^2 + a^3 + 
 
 + ^.) 
 
 (■("21 ^22 
 
 + 
 
 an 
 
 ^21 
 
 «12 
 
 «nia^l (^n2^2 
 
 ^2n 
 
 ^2» 
 
 a„t,x,. 
 
 + 
 
 ^21 1 ^22*^2 
 
 a2na;« 
 
 Notice that the coefficient of Xi in this sum is 
 
 ay.Au + a2iA2i + a.^Asi H h a,,f^„i = | ai„|. 
 
 9. As an application of the preceding, show that 
 
 2 (xi + a^a + a^s) 
 
 1 1 
 
GENERAL PROPERTIES OF DETERMINANTS. 
 
 X,' xi xi + 
 
 X^ X^ X., 
 
 + 
 
 Xo X^ Xi 
 
 x? xi x^- 
 
 
 1 1 1 
 
 1 1 1 
 
 
 X;X.2 XoX^ X^Xi 
 
 + 
 
 Xi X, X., 
 
 X.J X^ Xi 
 
 
 XiX.2 X2X^ X^Xi 
 
 1 1 1 
 
 
 1 1 1 
 
 1 . Given /^ (cc) = a^x" + 3 b^x- -f- 3 CiX + d^ , 
 fo{x) = ao-Tr 4- 3 b.^oi^-i- 3 c.^ -j- c?2, 
 Mx) = apy'-\- 3 b.x'-^ 3 c,x + d, ; 
 
 show that 
 
 /iW /I'W /i"W =-18 
 
 ^2^ f^'ix) f,"{x) 
 
 M^) fs\x) f,"{x) 
 
 I —X x^ — ar*^ 
 
 h bi Ci di 
 
 I2 O2 C2 CI2 
 
 h ^3 ^3 ^3 
 
 The first determinant is at once reducible to 
 
 -18 
 
 ciiX + bi biX + Ci CiX-i-di 
 agic-h^a &2^ + C2 C2X-\-d2 
 b^x + C3 Cgic + da 
 
 0-30; + 63 
 which may be written 
 
 10 
 
 «! a^x + bi biX-\-Ci CiOJ + c?! 
 
 «3 a^x + b^ hx-\-c.i c^x-\-d^ 
 
 Again using 37, the last determinant becomes the result above 
 written. The student's attention is called to the fact that the 
 method of bordering a determinant, z.e., increasing its degree 
 without changing its value, here employed, is frequently of use 
 in simplifying. 
 
 63. The following examples comprise several interesting 
 expansions of determinants. The cases considered and the 
 methods employed are important. 
 
 I. Expand the following determinant in ascending powers 
 of xi 
 
 X 
 
 a 
 
 
72 
 
 THEORY OF DETERMINANTS. 
 
 A is evidently a function of x of the nth degree, in which the 
 coeflacient of x"" is 1, and the absolute term is /(0) = |ai„|. 
 To complete the expansion, we have to find the coefficient of a^. 
 Consider the product of two complementary minors of A, of 
 the feth and {n — k) th degrees respectively, 
 
 otee + a; 
 
 a,, + x 
 
 and 
 
 %„-\-x 
 
 This product contains the term 
 
 x' 
 
 = ^n. 
 
 say. 
 
 The entire coefficient of ic* is accordingly SDn-*, *.e., the 
 sum of all the minors of |ai„I of order 7i—k, whose principal 
 diagonal lies in the principal diagonal of lai„|. 
 
 As an illustration, the student may show that 
 CLi-{-x bi Ci di 
 
 «3 ^3 C3 + a; ds 
 
 a^ 64 C4 d4-\-x 
 
 + \b2d^\-h\aM + \cM^x' 
 
 For another exercise, let the student find the terms of 
 A = |ai„| that contain k elements from the principal diagonal, by 
 considering the product of two complementary minors, as above. 
 
GENERAL PROPERTIES OF DETERMINANTS. 
 
 73 
 
 II. Expand 
 
 ax + ly CiX 4- n^y 
 
 h^ -f m^ a2X + ^22/ 
 in ascending powers of x and y. 
 
 Putting first y = 0, and then a; 
 and 2/^, respective!}*, are 
 
 \x + m^ 
 a^x 4- /i2/ 
 ca; -\-ny 
 
 0, the terms involving a^ 
 
 a ci &i 
 
 , and f 
 
 I rii 
 
 Wi 
 
 Ca h a. 
 
 
 ?i2 m 
 
 h 
 
 &2 <^*2 C 
 
 
 wig ^2 
 
 n 
 
 Putting the y's in the two last columns of A equal to zero, 
 we obtain for one set of terms involving afy 
 
 3?y 
 
 I 
 
 Cl 
 
 W2 
 
 h 
 
 m2 
 
 <X2 
 
 and the two other sets of terms containing x^y are, similarly, 
 a?y 
 
 a Ui bi 
 
 and a^y 
 
 a Ci mi 
 
 C2 m ai 
 
 
 C2 b Zi 
 
 &2 h c 
 
 
 &2 «2 ^ 
 
 The coefficient of xy^ is found in a similar manner, and the 
 entire expansion is accordingly 
 
 \=:X' 
 
 a 
 
 Ci &1 
 
 + 0^22/ 
 
 
 I c. 
 
 M 
 
 + 
 
 a Ui 
 
 &i 
 
 + 
 
 a 
 
 Ci mi 
 
 
 
 C2 b a. 
 
 
 
 712 6 tti 
 
 
 C2 m tti 
 
 
 Ca 6 ^1 
 
 
 
 62 ttg c 
 
 
 _ 
 
 ma Ota C 
 
 
 62 ^a c 
 
 
 62 ^3 n 
 
 — 
 
 4-a^ 
 
 
 a Wj mj 
 C2 m ?! 
 
 + 
 
 Z Ci mj 
 W2 & ?i 
 
 + 
 
 ? Wi &1 
 
 rig m «! 
 
 
 + 2/^ 
 
 I nj mi 
 na ?>i Zi 
 
 
 _ 
 
 ^2 ?2 
 
 n 
 
 
 WI2 
 
 ^2 ^ 
 
 
 m 
 
 2 ?2 c 
 
 
 
 
 ma k 
 
 n\ 
 
 III. Show that any determinant A may be developed in terms 
 of the elements of any row and column and the second minors 
 of A corresponding to the product of these elements. 
 
 Let A'=|rtii ^22 agsh 
 
 and border it as indicated below ; calling the result A, we may 
 
74 
 
 THEORY OF DETERMINANTS. 
 
 expand A in terms of the bordering elements and first minors 
 of A', i.e., 
 
 A= a^n am am «„« = ctoo A'— J aio ttoi Ai+«io «02 ^12 
 
 4-«10 <^03 ^13H-«20 «01 ^21+<^20 «02 -^22 
 "i a^ a^Q -^23 "I ^30 ^01 -^31T" ^30 <^ -"32 
 
 -f- ago ttos -^^as 1 1 
 
 in which Ai^^ is, as usual, a first minor (with its proper sign) 
 of A'. 
 
 ttoo 
 
 «01 
 
 «02 
 
 ^03 
 
 «10 
 
 «11 
 
 «12 
 
 ai3 
 
 «20 
 
 ^21 
 
 ^22 
 
 «23 
 
 ^30 
 
 ^31 
 
 «S2 
 
 ^33 
 
 In general, if A'= 1 a^ 022 •■ 
 
 , we have 
 
 aoo ttoi «02 ••• 
 
 aon 
 
 ft^Q ail %2 ••• 
 
 au 
 
 (7^0 ^21 «22 ••• 
 
 «2« 
 
 «nO«nl «n2"- 
 
 ^*nn 
 
 = aoo A'— 2 afoaot^i* (/, ^• = 1, 2, 3 ...w) , 
 
 in which, as before, Aij, is a minor of A'. 
 
 For .the terms of A containing a^ are obviously aooA'. Now 
 let (7 be the complementary niinor of 
 
 cioo OoA in A ; 
 
 then ttooCtiAC' contains all the terms of A involving Oooaf*; hence 
 afjO contains all the terms of A' involving (x,^, and consequently 
 
 and — ai^aQj^Aij, is the expression for the terms of A containing 
 the bordering elements a^o, ao** 
 
 This expansion, known as Cauchy's Theorem, is frequently 
 written 
 
 A ='«,, A,- :Sa,,a,,/?i,. (a) 
 
 Here A is a determinant of the nth order. A^, is, as usual, the 
 complementary minor of a^, in A ; i has all integral values from 
 1 to n, except r ; Tc has all integral values from 1 to w, except s ; 
 and ^ik is the complementary' minor of a,j in A,,. (a) is, 
 accordingly, the expansion of A in terms of the elements of the 
 rth row and the sth column. 
 
GENERAL PROPERTIES OF DETERMINANTS. 
 
 75' 
 
 The student may show that 
 
 A = 
 
 A = 
 
 a f g h 
 
 A b 
 
 9i ,0 c 
 
 hj^ d 
 
 ai-^Xi ag ag 
 
 = abed — Jficd — ggibd — hh^c. 
 
 
 
 a^ 
 
 
 
 —x^ Xq 
 — ajg x^ 
 
 ai+iTi ttg ttg a^ 
 
 — Xi ct'a 
 
 — Xl iCg 
 
 a;^ 
 
 -x"i 
 
 f ai Og ag a4 "I 
 
 = XiXoXoX, < i-\ — Y 
 
 i ^ J 4 j^ ' iCi i»2 ^3 ^4 j 
 
 IV. If A 
 
 and if we put 
 
 Xl ag ttg . 
 
 .. a„ 
 
 ai X2 ttg . 
 
 . . a„ 
 
 tti ag ajg . 
 
 •• '^n 
 
 «! as ttg . 
 
 .. a;„ 
 
 f{x) = (x^— ai) (iK2- 02) ••• (X,,- aj, 
 
 and 
 
 aa;. 
 
 we find 
 For 
 A = 
 
 A=f(x) + :S,aJ'(x,). 
 
 1 
 
 
 
 ... 
 
 1 
 
 Xl a.2 Og 
 
 ... a„ 
 
 1 
 
 tti a*2 ag 
 
 • • • Ct/t 
 
 1 
 
 ttj Oo iTg 
 
 ... OLn 
 
 1 
 
 ttj as ag 
 
 ... a;. 
 
 1 fti a2 ttg 
 
 la.'i— ai " 
 1 X2-a2 
 
 10 ajg— ag 
 
 1 0* 
 
 
 
 
 
 X^ — a^ 
 
 whence (if, as in III., we let A' represent the complementar}' 
 minor of the first element) A'=/(a!), and, since every first 
 minor of A' vanishes except the minors of the diagonal elements, 
 we have the required value of A on applying the theorem 
 
76 
 
 THEORY OF DETERMINANTS. 
 
 V. Show that 
 
 A = 
 
 CLi >C X OS , , , QC 
 X 02 X X ... X 
 X X a^ X ... X 
 
 = f{x)-xf{x). 
 
 X X X X ... a„ 
 
 in which f{x) = (x — aj) {x — ag) (a; — ag) ... (a; — a„) , 
 and f{x) = J^ = {x- a,,) (a; - ag) ... (» - a„) 
 
 + {X- ai) {x - ag) ••• (a; - a„) 
 
 + "'+(x-a,){x-a2)-"(x- a„_i) . 
 
 A= 10 ... = 1 — a; — x — a; 
 
 Itti— a; 
 1 ag-a; 
 10 ttg-a; 
 
 1 
 
 
 
 0... 
 
 
 
 1 
 
 ai 
 
 X X ... 
 
 X 
 
 1 
 
 X 
 
 a2 X ... 
 
 X 
 
 1 
 
 X 
 
 X ttg ... 
 
 X 
 
 1 
 
 X 
 
 X X ... 
 
 a„ 
 
 — X 
 
 
 
 
 
 a,-X 
 
 10 
 Then, as in the preceding example, 
 
 A=/(a^)-<(a:). 
 
 64. To the expansions of the preceding article we append the 
 solutions of the following determinant equations. 
 
 I. Solve the equation 
 
 A = 
 
 X 
 
 tti 
 
 «! 
 
 tti 
 
 «! 
 
 X 
 
 ttl 
 
 tti 
 
 tti 
 
 ai 
 
 X 
 
 ai 
 
 «! 
 
 tti 
 
 tti 
 
 X 
 
 = 0. 
 
 We find by easy reductions 
 
 A=(a;-aiy 
 
 X 
 
 tti 
 
 «! 
 
 «! 
 
 -1 
 
 1 
 
 
 
 
 
 -1 
 
 
 
 1 
 
 
 
 -1 
 
 
 
 
 
 1 
 
 = (a;-ai)3(a;4-3ai) = 0. 
 
 Whence, 
 
 a; = ai, ai, ttj, —3 tti. 
 
GENERAL PROPERTIES OF DETERMINANTS. 
 
 77 
 
 II. Find the values of x in the equation 
 
 A = 
 
 X 
 
 ai 
 
 h 
 
 C] 
 
 ttl 
 
 X 
 
 Ci 
 
 &1 
 
 h 
 
 Ci 
 
 X 
 
 a. 
 
 Ci 
 
 h 
 
 «1 
 
 X 
 
 = 0. 
 
 A = 
 
 a;+«i+^+Ci «! 61 Ci 
 ar+ai+6i+Ci a; Cj 61 
 
 x-i-ai-\-bi-\-Ci 61 ai a; 
 
 = (a;-f ai+6i+Ci) 
 
 1 
 1 
 
 1 
 
 1 
 
 X 
 
 &1 
 
 X < 
 
 ^1 
 
 = (a; + ai+ &i+ Ci) (a; — ai+ 61— Ci) 
 
 -1 1 
 
 1 X Ci 
 
 -1 
 
 • 
 
 
 1 Ci X 
 
 1 bi «! 
 
 a; 
 
 
 Put the two polynomial factors =A and J5 respectively 
 e last expression 
 
 then 
 
 A.B. 
 
 0-10 
 
 1 x-hc^ Ci 61+ Ci 
 1 x-\-Ci X ai-{-x 
 1 6i-f-«i «i cii+a; 
 
 
 
 
 = 
 A.B. 
 
 
 
 1 
 1 
 
 1 feiH-Oi— a; — ( 
 
 -1 
 a; 
 
 
 
 -ai- 
 
 
 
 
 
 X 
 
 • 
 
 
 
 Whence 
 
 {x -\- «!+ 61+ Cl) (x — a^— Ci+ &i) (5i4- ai— a; — Ci) 
 (tti+aj — &i— Ci) = 0. 
 
 .-. a; = -(ai+6i+ci), (a^- ftj-j-Ci), (^^-Ci+ai), 
 (61-tti+Ci). 
 
 III. Find the roots of the equation 
 
 A = 
 
 = 0. 
 
 a^ W (? 
 
 {a-^xy {b^-xy {G+xy 
 {2a + xy {2b+xy {2c-\-xy 
 
 From the third row of A subtract the first row multiplied by 8, 
 
78 
 
 THEORY OF DETERMINANTS. 
 
 aud from the second row subtract the first row. Then subtract 
 the second row from the third, and we have 
 
 = 3X2 
 
 o? V" <? 
 
 3a2 + 3aA. + A2 ZV^ + ^hX + X^ Z<^ + ZcX + X., 
 
 3a2-f-aX 362 + 6X Z(?-{-cX 
 
 Now subtract the third row from the second, and 
 
 A = 3\'^ 
 
 a^ h^ <? 
 
 2a +X 26-f-A. 2c+A 
 Sa^H-aA ?>W+hX Zc'-^cX 
 
 = 0. 
 
 From this equation it is obvious that three values of X are zero ; 
 the other two roots can be found by equating to zero the quad- 
 ratic factor of the first number, and solving for A.. 
 
 A ma}^ however, be further simplified as follows : subtract 
 the first column from each of the other two ; then 
 
 A==3\Hc-a)(6-(x) 
 
 2a-\-X 2 2 ■ 
 
 3tt2+aA. 36 + 3a+X 3c+3a4-A. 
 
 Now subtract the second column from the third, and 
 
 A=3\3(6-a) {c-a) (c-b) 
 
 a? a2-f-a6-f62 a-\-h + c 
 
 2a+X 2 
 
 3a2+a\ 3a + 364-X 3 
 
 Finally, add the second column multiplied by —a and the third 
 multiplied by ab to the first, and afterward subtract the third 
 multipUed by a + & from the second ; then 
 
 A=3X«(6-a) (c-a) (c-6) 
 
 ahc —bc — ca — ab a-\-b-\-c 
 X 2 
 
 A 3 
 
 0. 
 
 Whence three values of X are seen to be zero, aud the other 
 two roots are readilv found from the quadratic 
 
 (a + 6 + c) X'+ 3 (&c + ac + a2>) X + 6 abc = 0. 
 
GENERAL PBOPERTIES OF DETERMINANTS. 
 
 79 
 
 65. Theorem. — The total differential of a determinant A 
 is a sum of n determinants, each oftvhich is obtained from A by 
 substituting the differeyitials of the elements of a row for the 
 elements themselves. 
 
 Let A= \x^y2Z^---t^\. 
 Developing in terms of the elements of the ith. row, 
 A= x,X,-\- y,Y,-\- ;2^z,+ ... + t.T^. 
 .-. dA = dx,Xi+dyiYi+dZiZi-\ \-dt,Ti. 
 
 There must be n such expressions for the total differential, each 
 of which is obviously A, after changing the elements of the ^th 
 row into their differentials. 
 
 .-. [c2A]* = 
 
 + 
 
 From the differentials, partial or total, we, of course, pass 
 to the coiTesponding derivatives in the usual way. 
 
 Illustrations. 
 
 dxi dyi dzy 
 
 ^2 y-i % 
 
 ...dt, 
 
 ... t2 
 
 + 
 
 »i 2/1 % 
 dx2 dy2 dz2 
 
 ... t, 
 
 ,..dt2 
 
 ^n Vn ^n 
 
 ... t,, 
 
 
 ^u Vn ^n 
 
 ... t. 
 
 X^ 2/1 ^1 
 ^2 2/2 ^2 
 
 ... t, 
 ... t2. 
 
 • 
 
 
 r 
 
 dx^ dy^ dz^ 
 
 '.'.'. di 
 
 
 
 
 V — — ; N'^dv 
 
 N 
 
 dM M 
 
 dN N 
 
 . d 
 
 dM M 
 
 dN N 
 
 = 
 
 d'^M M 
 
 d'jsr N 
 
 Let 
 
 (^U ^22 ^33 ^44 1 
 
 a 26 
 
 c 
 
 a 
 
 26 c 
 
 b 2c 
 
 k 
 
 b 
 
 2c k 
 
 dA 
 
 da 
 
 J^ll-\-A22 — 
 
 a 26 c 
 
 + Jc 
 
 2c k 
 
 
 6 2c k 
 
 
 a c 
 6 k 
 
 The [ ] denote the total differential. 
 
80 
 
 THEORY OF DETERMINANTS. 
 
 ^ = 2A^+2A^+A,,-^A^=4b\l k 
 
 -4k 
 
 a b 
 b c 
 
 2b 
 a 
 b 
 
 c 
 
 26 
 2c 
 
 a c 
 b k 
 
 ■~7~ — -^13"f'-^24~f~ 2As2-}- 2A4Q = •••, 
 
 ac 
 
 dA 
 dk 
 66. Theorem 
 
 -^33 + -^44 
 
 dA 
 
 If the elements of A are all functions of the 
 
 same variable x, ^^— equals the sum of n determinants, each of 
 ux 
 
 which is obtained from A by substituting the derivatives of the 
 elements of a row for the elements themselves. 
 
 The truth of this proposition is evident from the preceding. 
 
 Thus, if 
 
 6A 
 dx 
 
 Mx) f22ix) . 
 /nl(aj) fn2{x) . 
 
 /2l(a5) /22(«) 
 
 f2n{x) 
 fnn(x) 
 
 JJix) 
 
 • Mx) 
 
 'fnnix) 
 
 + 
 
 /uW fu{x) .../in(a;) 
 fj(^\ f.J(r.\ fj(^a:) 
 
 + 
 
 If 
 
 1 Xiic 
 
 1 1 Aaic 
 
 1 1 X^x 
 
 1 1 
 
 
 
 fn'ix) f^\x) 
 
 fnl{x) fAx) 
 
 fn(x) f^(x) 
 U{x) f^x) 
 
 fnl'(x) fj(x) 
 = A1A.2X3 1 
 
 -fnnix) 
 
 'An{x) 
 'f2n{x) 
 
 Jnn'ix) 
 
 + ••• 
 
 - a; 
 X 
 
 1 1_ 
 A2 A2 
 1 1 
 
 1 
 
 
 
 — X 
 A, ^ 
 
GENERAL PROPERTIES OF DETERMINANTS. 
 
 81 
 
 the student may show that 
 
 dx 
 
 — — AiX2^S 
 
 1 
 
 X 
 
 ^3 
 
 1 1 
 
 + 
 
 — a; 
 
 ^ X 
 
 A3 
 
 1 
 
 + 
 
 1 
 
 X 
 
 
 
 
 Ai 
 
 
 
 
 1 
 
 1 
 
 
 
 
 
 X 
 
 
 A2 
 
 A2 
 
 
 
 
 
 
 
 1 
 
 
 "b 
 
CHAPTER III. 
 
 APPLICATIONS AND SPECIAL FORMS. 
 
 67. We have now discussed the origin and some of the 
 properties of determinants ; it remains to show how useful these 
 functions are in application, and to examine some of the Special 
 Forms that are of frequent occurrence. Within the limits of an 
 elementary work like this it will be possible to select only a 
 very few of the many important applications, and to touch 
 somewhat briefly upon the special forms. Enough will be given, 
 however, to enable the student to pursue his further investiga- 
 tions with pleasure and profit. We now return to the problem 
 with which we commenced the presentation of determinants, and 
 proceed to the 
 
 ^^ Solution of Linear Equations, and Elimination. 
 
 68. Consider the set of three simultaneous linear equations : 
 
 aiX + 6i2/ + CiZ = 7?ii 
 
 «1 
 
 h 
 
 Ci 
 
 a^ 
 
 h 
 
 C2 
 
 cts 
 
 h 
 
 C3 
 
 a<fc + b^y -\- c^ = ^2 [>■ , and A 
 a^x -i- b^ -^ c^z = 7ns 
 
 Multiply these equations by Ai, A^, and A^ respectively, and 
 add by columns, obtaining : 
 
 (aiAi+ a2A2+asAs)x + (61^1+62^4-^3^3)2/ 
 + (Ci^l + C2A +03^3)2 
 = miAi-\-m2A2-\-m^As. 
 
 By 45 the coefficients of y and z vanish ; the coefficient of x 
 
 is A = |ai62C3l, and the absolute term is I mi 62 C3 1. 
 
 Whence ^,_ \m^ b^ Cgl ^ 
 
 ki 62 C3I 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 83 
 
 y = 
 
 If we had multiplied the given equations by By^ -Bg, -B3, we 
 should have caused the coefficients of x and z to disappear in 
 the resulting equation, and would have found 
 
 1^1 ^2 C3I 
 
 Using (7i, C2, C3 as multipliers, we should find, similarly, 
 
 ^_ ki 62 ^3! , 
 ki 62 C3! 
 
 69. To generalize the solution of the preceding article is now 
 an easy step. Given 
 
 and 
 
 ^21^1 "I tt22*^2 "I • * * "T" tt2r'^r I 
 a,^X^ 4- «V2^2 H f- (^rrXr + 
 
 ■' + a2nX^ = m,2 
 
 ■ * "T" Clrn'^n = '^r 
 
 , I., 
 
 «11 
 
 «12 
 
 ... ai, . 
 
 • «ln 
 
 «21 
 
 ^22 
 
 .. ttgr . 
 
 . «2n 
 
 a,i 
 
 a,2 
 
 .. a,r . 
 
 . «,.« 
 
 am 
 
 «„2 
 
 .. ttnr . 
 
 . a„„ 
 
 Here A is, as before, the determinant formed from the n^ co- 
 efficients in the first members of equations I., and is called the 
 determinant of the system. 
 
 Multiplying equations I. in order by A^^^ A^r-, ... A„^ ... yl„^, 
 and adding by columns, we find 
 
 + OjriArr + * * * + f^nzAnr) ^2 
 
 I \^12Air -f- (^22A2r ~\~ ' ' 
 + ... 
 
 + 
 
 + a„,^,rH f- a„,,A^,)x^ 
 
 -\-m,A„-\ Vm^A^,. 
 
 (^) 
 
84 
 
 THEORY OF DETERMINANTS. 
 
 In equation (A) the coefficient of all the unknowns except the 
 coefficient of x^ vanish, and the coefficient of x^ is obviously A. 
 The second member of (A) is evidently what A becomes when 
 mi, mg, ...m^ are put for the corresponding elements of the rth 
 column. Hence 
 
 ttll 
 
 a,2 
 
 
 mi 
 
 ... tti. 
 
 ^21 
 
 ^22 
 
 
 Wg 
 
 ... ^2. 
 
 ftrl 
 
 a.2 
 
 
 m. 
 
 ... a,„ 
 
 am 
 
 (^n2 
 
 
 w« 
 
 ... a„„ 
 
 
 ttn ai2 .. 
 
 . «!, . 
 
 . a,. 
 
 
 ^21 ^22 . . 
 
 aa, . 
 
 . a2« 
 
 -^ 
 
 a,.i ay2 .. 
 
 arr . 
 
 . a^ 
 
 
 a„i a„2 .. 
 
 a„, . 
 
 . a«n 
 
 Translating this formula, we have : 
 
 The value of each of n unknowns in a set of n linear simul- 
 taneous equations is the quotient of two determinants; the divisor^ 
 (denominator) is the same for all the unknowns and is the deter- 
 minant A of the 7ith degree formed by writing the coefficients of 
 the unknowns in order (i.e., the determinant of the system) ; the 
 numerator of the value of any unknown as x^ is obtained from A 
 by substituting for the elements of its rth column the second mem- 
 bers of the given equations in order.* 
 
 70. The following modification of tho solution already given 
 of equations I. will be interesting. Employing the same notation 
 as in 69, we have 
 
 flj^A 
 
 which, by 37, 
 
 an 
 
 a^2 • 
 
 ^21 
 
 a^ . 
 
 ... 
 
 ... . 
 
 a,i 
 
 a.2 . 
 
 a„i 
 
 a«2 • 
 
 airXr 
 
 
 a^rXf 
 
 
 ... 
 
 
 a^x. 
 
 
 ... 
 
 
 a«r»r 
 
 
 «1« 
 
 a«« 
 
 * This is the rule for the solution of simultaneous linear equations first 
 obtained by Leibnitz, and subsequently rediscovered by Cramer. (See 
 opening paragraph of Chapter I.) 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 85 
 
 
 ^2r-l ^21'^l~r ^22*^2 1" 
 -f- a2n ^» C(-2r+i 
 
 • • "f" Cllr-l^r-l'T' (^Ir-^rl 
 " + Ot2r-l^r-l~H Cl2r^r~\~ 
 
 " + arr-lX,_i-^ a,rXr + 
 
 " + anr-lXr-i+ar,rXr-\- 
 
 <^2/i 
 
 Now substitute in the last determinant the values of the 
 elements of the rth column, and 
 
 a? A = 
 
 Oil ai2 . . . 
 
 ^21 ^22 • • ' 
 
 Ot,,i Cl„ 
 
 «2r-l 
 
 
 «1« 
 a2n 
 
 a,,_i m. 
 
 m. 
 
 .. a„ 
 
 
 I Cll\(^22 • • • ^rr • • • ^nn ' 
 
 as before. 
 
 A simple example of the methods of 69 and 70 is the solution 
 of the following equations : 
 
 .'. X 
 
 2x + 6y — 3z=lS 
 Sx-Sy-\-2z = 21 
 
 48 3 3 
 18 6-3 
 21-3 2 
 
 Here A = 
 
 5 3 3 
 2 6-3 
 8-3 2 
 
 = -231, 
 
 -231 
 
 3; 2/ = 
 
 5 48 
 
 3 
 
 2 18- 
 
 -3 
 
 8 21 
 
 2 
 
 -231 
 
 5: z= 
 
 5 3 48 
 2 6 18 
 8-3 21 
 
 231 
 
 = 6. 
 
 As another example, we may solve the equations : 
 
 z H-w +flj = b 
 ti-{-x -\-y = c 
 ^ -\-y +2J =d 
 
 Here A = 
 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 = -3. 
 
86 
 
 THEORY OF DETERMINANTS. 
 
 The student may show that 
 
 a; = i(6+c-f d-2a); 2/ = J(c +d + a- 26) ; 
 z=l(d-{-a-\-b-2c); u = l{a-\'b -^c -2d). 
 
 71. We have hitherto tacitly assumed that neither A nor m< 
 (i= 1, 2, ... 7j) should vanish. If A vanishes and m^ does not, 
 the value of each unknown becomes infinite. If m^ vanishes 
 while A does not, the values of the unknowns are severally zero ; 
 but when mj vanishes, the system consists of homogeneous 
 equations, and their solution is given later. If m^ does not van- 
 ish, but A and the numerators of the unknowns do vanish, then 
 we have the following theorem. 
 
 72. IftJie equations of a set are not independant, i.e., if any 
 one {or more) is a consequence of the others, the value of each 
 
 imknown takes the form — 
 
 Since the equations are all linear, any one can be derived from 
 the others only by the addition of two or more of them after 
 each has been multiplied by some constant factor. But this 
 gives rise in the determinant numerator and denominator of the 
 value of any unknown to two or more identical rows, and hence 
 numerator and denominator vanish. 
 
 For an example, take 
 
 aiXi-[-biX2 +CiXs = mi 
 OoCCj + 62^2 + C2a;3 = '^h 
 aiXi-\- aibiX2+ aiC^x^^ aimi 
 We find 
 
 h &i Ci 
 
 where A = a. 
 
 Xi= tti 
 
 ^2 &2 ^2 
 
 mi 61 Ci 
 
 = 5; a., = ai'^^-^ = 5; x,^ 
 A 
 
 «! 61 Ci 
 
 ao &2 ^2 
 ttj bi Ci 
 
 \ai &2^%I 
 
 0. 
 
 For a second example, the student may show that the values 
 of the unknowns in the following equations take the form -• 
 
 4 
 
 3a; + 2?/ — 02 = 
 6a; — 3?/ -f- 4z 
 y-2z 
 
 = 4^ 
 
 = 22 L 
 = -2J 
 
APPLICATIONS AND SPECIAL FORMS. 87 
 
 m^_i = 0, and one m as m^ does not, 
 
 ^n^nn 
 
 73. If mi = W2== •• 
 we evidently get 
 
 _ m^A,a 
 
 Whence J^ = ^= ... -|!L = !??l\ 
 
 74. If mi = m2= ••• =m„=0, ^.e., i/* equations I. become 
 homogeneous, tJien, unless x^, x^^ •••^« «»*e severally zero, A mi^s^ 
 
 In that case, equations I. become 
 
 ?! = <Xn^i+ tti2CC2 + 
 
 + a2^ir,.+ 
 
 + «2Ha^n=0 
 
 (m^ being zero), the 
 
 II. 
 
 l^ = arl^l+ «r2^2H h «rr^% + 
 
 Since a:^A = I an a22 <^33 • • • 'wi,. ••• <^nnl 
 truth of the assertion is obvious. 
 
 An example is furnished by the homogeneous equations : 
 
 an a^i -f ai2 if2 4- «i3 a?3 = ^ 
 
 a2ia?l + «22^2 + «23»^3-=0 I. {E) 
 
 ctsiX^ + a32a^2 + «33a^3 = J 
 
 Multiplying equations (jEJ) by A^, A^i, ^31, respectively, and 
 adding by columns, we have 
 
 (an^n + «2i^2i + «3i^3i)a^i 
 
 + («12^11 + «22^21 + a32^3l)^2 
 
 4- (ai3^ii + a-23^21 + 033^31) a's = 0. 
 The coefficients of X2 and x^ are zero, and we have 
 
 a;iA = 0, .-. A = 0. 
 
 As a further illustration, the student may show that if 
 
 nXiX-\-vyiy+icz^z-\-Ui{yz^+yiz) -i-Vj^izXi-^z^x) -\-w^{xy^+x^y) 
 is zero for all values of x, y, and z, then 
 
 uvw — uui— vv^— wWi-\- 2 ?/ii'iWi= 0. 
 
88 
 
 THEOKY OF DETERMINANTS. 
 
 Observe that by the given conditions the coefficients of a;, 
 ?/, z must severally vanish. 
 
 75. With the help of 74 we obtain an interesting proof of 
 the multiplication theorem of 53. Consider the simultaneous 
 equations 
 
 (ai-X)a;i-t- h^x^ + qajg =0^ 
 as^i +(&2 — A)a72+ Cg^s =0 > I. 
 (h^ + h^2 +(C3-X)a;3=0 ) 
 
 By the preceding article we must have 
 
 «! — A 6i Ci 
 
 CI2 62 — •^ C2 
 
 «3 &3 Cg — X 
 
 or 
 
 = 0, 
 
 0; 
 
 (a) 
 
 X^-Jfx^ + iVA-P 
 where we notice especially that 
 
 P = I «! 62 C3 I . 
 
 Let the roots of (a) be Ai, Ag, A3 ; then, evidently, 
 
 P = — Ai A2 A3. 
 
 Now, from I. we obtain three new equations as follows : 
 Multiply equations I. b}^ aj, as, ag respectively, and add them 
 together ; also multiply equations I. by ft, ft, ft respectively, 
 and add ; finally, multiply equations I. by yi, yg? 73 respec- 
 tively, and add. We now have three new equations where the 
 determinant of the S3'stem is 
 
 A'= aiai + «2a2 + C^3a3 — ttiA ftittj + &2a2 + ?>3a3 — a^A 
 
 C^lft+ «2ft+ <X3ft- ftA 61ft + 62ft + ?>3ft- ftA 
 
 «i7i + 0^2 72 + «3 73 — 71^ &i7i + ^272 + &3 73 — 72^ 
 
 Cittj -}- C2a2 4- Cgag — ttgA 
 
 Cift+C2ftH-C3ft-ftA 
 Ci7i + C272 + C3y3 — ygA 
 
 0, 
 
 or 
 
 QA3-JIfiA2 + iViA-Pi = 0, 
 
 (&) 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 89 
 
 where we observe that Pi is what A' becomes when we put 
 X = 0, and that 
 
 Q = lai ^2 ysl* 
 Further, since 
 
 it follows that 
 
 — — — A.1 Ag Ag — P, 
 
 Pi = PQ= Ui &2 C3I X Ui ^2 ysl- 
 
 But Pi is exactly the determinant obtained by 53, and this 
 was to be shown. 
 
 76. The condition A = being fulfilled, the equations no 
 longer determine the actual values of the unknowns ; they deter- 
 mine only the ratios of these values. For, if a;/, x^-, ...xj 
 satisfy equations II., so will Jcxi, kx^j... 'kxj, k being any factor. 
 Any n — 1 of the given equations will suffice in general to de- 
 termine the ratios of n~l of the unknowns to the remaining 
 one. An example will make this clear. We employ for brevity 
 only three equations : 
 
 aiX-{-biy -|-Ci2; = 
 a2X -\-b2y -{-C2Z = 
 a^x -{- b^y + CqZ = 
 
 Write these equations 
 
 X 
 
 (a), 
 
 tti - 4- Ci - 
 
 y y 
 
 b. 
 
 X z 
 
 [if>)' 
 
 X . z 
 
 «3-+C3- 
 
 -b. 
 
 From any two of equations (b) we may find the values of 
 
 X z 
 
 - ; thus from the first two 
 
 y y 
 
 \bi Cgl . z 
 
 1% Col ' y 
 
 \cti &2I 
 |ai C2I 
 
90 THEORY OF DETERMINANTS. 
 
 Again, equations (6) are to be simultaneous ; hence these 
 lues of - an( 
 
 y 
 
 Substituting, 
 
 dC z ' ' 
 
 values of - and - must satisf}^ the third equation 
 
 ttgl&i C2I — ftglai Col + Cgltti 62I = ; 
 or A = 0. 
 
 Since from the preceding equations - also equals 
 
 y 
 
 l&i C3I I&2 C3I 
 
 and hence 
 
 or, 
 
 Idi Csl las C3I 
 
 we have ^ — Al — -^ — -^ 
 
 y" Br B~ b; 
 In the same way, 
 
 z _ C^ _ C2 __ C3 
 
 y B, B, b; 
 
 z Oi C2 O3 
 
 x:y:z = Ai:Bi:Ci 
 = ^2 • B2 ' G2 
 = ^3:^3:03. ' 
 
 That is to say. The ratio of any two unknowns in a set of 
 homogeneous equations is equal to the ratio of the cofactors in A 
 of the coefficients of these unknowns in any of the given equations. 
 
 The general proof of the proposition just stated may be given 
 as follows. "We have to show (equations II.) that 
 
 Xi'.x^'.Xs: ••• : x^ : .•• : x^^Au : ^12 : -i4i3 : .•• : ^1, : ••• : A^^ 
 = A21 : -^22 : A2S : '" : A^r : • • • : Am 
 
 ^=^ A^\\An2''A^'. "*'. A^y. *.'"'. Ann* 
 
 If these proportions are true, we must have the equations 
 x^=\Aj^ (X = constant; p = 1, 2, ••• n). (^1) 
 
APPLICATIONS AND SPECIAL FORMS. 91 
 
 The equation ^ 
 
 is always true, whatever the value of p, since A is itself zero. 
 Substituting in (^2) the values of 
 
 as obtained from (J5Ji), and multiplying by X, there results 
 a.iXi -f a,2a;2 + a.3^3 H h (^rr^r H h ^rna^n = 0. 
 
 This last being a true equation, the proportions from which it is 
 derived must hold.* 
 
 /■.-/^'• 
 
 77. From the last article, or the two preceding articles, 
 we deduce the important conclusion. In order that n linear 
 homogeneous equations may be simultaneous, it is necessary and 
 sufficient that the determinant of the system vanishes. In that 
 case any one of the equations is expressible linearly in terms of 
 all the others, provided the first minors A^^ do not all vanish. 
 For we have in general, A being zero, and Zi, Zg, ...Z„ repre- 
 senting the linear functions of equations II., 
 
 hAiie + hAzk + • • • + ^nAnk = ; 
 
 hence, if one at least of the first minors Au,, A2U, . . . A^^ is not zero, 
 as for example A^^, l^ must be expressible linearly in terms of 
 /g, ?3, '"In-) and hence Zi = is superfluous. If all the first 
 minors vanish, and one at least of the second minors does not, 
 then, similarly, it may be shown that two equations are super- 
 fluous, the system being doubly indeterminate, and so on. 
 
 78. Among the proportions of article 76 consider the 
 following : 
 
 Xi'. X2'. x^: ••• Xn =-4„i : An2 : A^ : ••• Anr^. (P) 
 
 * This demonstration applies of course so long as tlie first minors of A 
 do not all vanish. V 
 
92 THEORY OF DETERMINANTS. 
 
 ^ni5 ^«2i ^«3i '•'^nn ^16, nooe of them, functions of the 
 coefficients of the last equation of set II. in 74, 
 
 Hence, proportions (P) give the ratios of the unknowns 
 
 ajj, iCg, a^a, ••• a;„, that satisfy the n — 1 equations 
 
 a^iXi -{-ai2X2 -i \-auX^ ^^1 
 
 a2iXi -\-a22X2 -\ \-a2„x,, =0 liii.^ 
 
 if we denote by A^i the determinant formed from the co- 
 efficients in equations III. after suppressing the first column of 
 terms, by ^„2 the determinant formed from the coefficients of 
 equations III. after suppressing the second cohimn of terms, 
 and so on. Hence having given n homogeneous equations con- 
 taining n + 1 unknowns 
 
 anXj^-\-ai2X2-\ l-ai«+i«n+i = "^ 
 
 a2ii»i 4- «22^2 -\ f- (^2n+lXn+l = I jy^^ " 
 
 aniXi-{-a„2X2-] h«nn+ia^'«+i = J 
 
 we find the ratios of the unknowns as follows : 
 
 put A, = (-iy 
 
 (Xjl ai2 '•' «i,_i CLii-^i ••• «l»-fi 
 ^21 ^22 *'• ^2J-1 ^'2i+l *•• ^2n+l 
 
 Then from what precedes 
 
 x^: X2: x^: '" : x^T^ = di,: ^2' ^3' '" ' A«+i- 
 
 79. Consider the following n equations containing w — 1 
 unknowns. 
 
 aiiXi +ai2X2 H hoT'in-i^'i +i>i =0 
 
 «2ia^l +a22a?2 H |-«2n-iaJn-l +i?2 =0 
 
 dn-llXl-h O.n-1^2 H h a«-ln-iaJn-l + Pn-l= 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 93 
 
 Equations V. may be made homogeneous by multiplying them 
 by u, and regarding XiU, X2U, ...x^u^ u, as the unknowns, u 
 being any arbitrary quantity. Whence, if these equations are 
 simultaneous, we have by 77 
 
 = 0. 
 
 On 
 
 ai2 ... am-i 
 
 Pi 
 
 ^21 
 
 a22 ... a2n-i 
 
 P2 
 
 ... 
 
 ... ... 
 
 ... 
 
 CTn-n 
 
 Gt'n-12 . . . Ctn-ln-1 
 
 Pn- 
 
 am 
 
 a«2 ... (^nn-1 
 
 Pn 
 
 This result may be expressed as follows : n equations (not 
 homogeneous) containing n — 1 unknoicns are simultaneous if the 
 determinant of the nth degree formed from alt the coefficients 
 (the second members of the equations being included among these 
 coefficients) vanishes. 
 
 This condition could also be derived from equations II., Art. 
 74, by putting ic^=l. Those equations, n in number, then 
 contain n — 1 unknowns ; and if the equations are simultaneous, 
 we see that lai^l must vanish. 
 
 80. With the help of the preceding article another solution 
 of a set of linear equations may be obtained. For brevity we 
 employ only three equations : 
 
 (1) aiXi-{-biX2-^CiXs=mi 
 
 (Z) a2Xi -\- O2X2 -J- e2X^ 
 
 (o) a^Xi + b'iX2 -\- CqXq — iivQ. 
 
 Take with these equations another, 
 
 (4) o^ Xi -\-b^X2-\- C4 Xs = m^y 
 
 which we suppose consistent with the first three, and in which 
 ^^4? &49 C4, m^ are undetermined. By 79 
 
 = ma L 
 = mnJ 
 
 or, 
 
 where, as usual, 
 
 \ai 62 C3 m^\ = 0; 
 a^A^ -f 64 ^4 -f- cJJ^ + W4 Jf4 = ; 
 
 A^ = — \bi C2 mgl ; ^4 = I aj Cg mgl ; 
 C4 = — !«! 62^31 ; Jf4=Iai 62^3! = ^. 
 
 (5) 
 
94 THEORY OF DETERMINANTS. 
 
 Now if we eliminate m^ from equations (4) and (5), we get 
 
 Since equation (6) must be true whatever the values of a^, 64, C4, 
 the coefficients of a^, 64, c^ severally vanish. 
 
 A, 
 
 
 fl?2 
 
 __:?4. 
 
 or, 
 
 _ \mi 62 C3I . _ \ai mo cj 
 
 Xi — -^ ■ ; Xo — -= 
 
 A A 
 
 O4. 
 
 "aT ' 
 
 81. Let us now return to equations I. Art. 69. Considering 
 mi, ma, mg, ... m^ as linear functions of the ic's, we can express 
 any new linear function 
 
 CiXi + C2X2 H h c,ifl;„ = 2/ 
 
 in terms of the m's. 
 Thus, if we have given 
 
 by 79, 
 
 aiiXi-}-ai2X2 + 
 a2ii»i4-a22^*2 + 
 
 A' = 
 
 • -i-CnXr, =2/ ^ 
 
 • + CllnXn = Wi I 
 
 or 
 
 Ci C2 ... c„ 2/ 
 «!! ai2 ... «!„ mi 
 
 a2i a22 ... Cl2n ^2 
 
 a„i a„3 ... a„„ m„ 
 Now if A = Itti,,! , we readily obtain 
 A'±2/A = ±2/A; 
 Ci C2 ... c„ 
 
 + a,.«a;n = m„ 
 = 0. 
 
 ±2/A = 
 
 an aj2 
 
 ^21 <^22 
 
 a2„ m2 
 a„„ m„ 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 96 
 
 82. We have seen that if 7i homogeneous equations are to be 
 consistent with each other (simultaneous), the determinant of 
 the system must vanish. The equation 
 
 A = 
 
 then is an equation of relation between the coefficients, and is 
 really the result of eliminating the unknowns from the given 
 equations. We shall soon investigate this resulting equation of 
 condition or resultant in detail. We here deduce a general 
 form by which the result of eliminating n unknowns from p 
 given Imear equations, supposed simultaneous, may be ex- 
 pressed, 2^ being greater than n. 
 Given 
 
 VII, 
 
 (hi^i -{-a^X2-\-" 
 
 . +ai«a;„ =0 
 
 a2iXi-i-a.22X2-{-" 
 
 + «2n^« = 
 
 «nia^i + a„2«^2+" 
 
 .+a„„a;„=0 
 
 aj,iXi + aj,2X2 + ... + Sn«n = 
 
 If these equations are to be satisfied for other than zero val- 
 ues of the variables, the determinant of the system for any n 
 of them must vanish by 77. The equation expressing this con- 
 dition is obtained by writing 
 
 an . ai2 
 
 ^^21 <^22 
 
 0. 
 
 (M) 
 
 Equation (Jf ) is accordingly interpreted to mean that every 
 determinant of the nth order formed from any n rows of the 
 matrix on the left must vanish. For an example the student 
 may eliminate the two ratios a?! : iCg : x^ from the five equations 
 
 aiXi + biX2-\-CiX^ = (i= 1, 2,...5), 
 
96 
 
 THEOEY OF DETERMINANTS. 
 
 obtaining the equation 
 
 «! 
 
 h 
 
 Cl 
 
 ^2 
 
 h 
 
 C2 
 
 as 
 
 h 
 
 C3 
 
 a^ 
 
 h 
 
 C4 
 
 as 
 
 0, or 
 
 83. Suppose we have given 
 
 ttl 
 
 as 
 
 ttg 
 
 a4 
 
 «5 
 
 &1 
 
 &2 
 
 2>3 
 
 &4 
 
 h 
 
 Ci 
 
 C2 
 
 C3 
 
 C4 
 
 Cs 
 
 then, by 78, 
 
 
 c^Xs + d^x^^ 
 C2CC3 + d^^ 
 c^Xs + d^^ 
 
 = 0, 
 = 0, 
 = 0; 
 
 Xi'.x^'.x^: 
 
 i»4=l^i C2 ds\ : — 
 
 laj^c^dsl : 
 
 kiJ 
 
 Substituting 
 
 the values of 
 
 
 
 
 x^ x^ 
 
 ■ — •) — "i 
 
 Xs Xs 
 
 i»4 
 
 — ? 
 
 
 l«i h dsl ' —1% 62C3I 
 
 we get the relations 
 
 61 1 ai C2 c?3 1 + Ci I «! &2 c?3 1 — (^1 1 ai 62 
 
 «i 1 ^1 C2 ds 
 
 a2 1 61 C2 C^3 1 — &2 1 <^l C2 C?3 1 + C2 I «! &2 <^3 I 
 «3 I &1 C2 C^s 1 — &3 1 tti C2 ^3 1 + C3 1 tti 62 C^3 I 
 
 which are all expressed by the matrix 
 
 0, 
 
 C?2 I tti 62 C3 1 = 0, 
 
 ds I »! 62 C3 1 = 0, 
 
 ai 
 
 h 
 
 Ci 
 
 ^1 
 
 a^ 
 
 h 
 
 C2 
 
 d2 
 
 as 
 
 h 
 
 Cs 
 
 ds 
 
 To generalize this, we return to art. 78. 
 From equations IV. we found 
 
 x^'.x^iXs'.'" a;„+i = Ai : Ag : A3 ••. A„+i. 
 
 Substituting in equations IV. the values of 
 
 — •> 
 
 X, 
 
 Wi 
 
APPLICATIONS AND SPECIAL FORMS, 
 given by these proportions, we have 
 
 ttiiAi 4- aio^a -\ f- (^hr X-\ f- «ln+l ^«+i — 1 
 
 ttsiAi + a.22^2 H 1- «'2,- A, H h cf2H+i A^+i = 
 
 a.iAi + a,2A2 H f- a,. A, H \- a,„+i A„^i = 
 
 a„iAi -f a„2A2 H h «»rA, H f- a^n+An+i = J 
 
 These n relations are expressed by the matrix 
 
 97 
 
 (li) 
 
 W'21 ^22 
 
 an a,2 
 
 «2r 
 
 ttln 
 
 «ln+l 
 
 a2« 
 
 «2«+l 
 
 ... 
 
 ... 
 
 «,-n 
 
 ttm+l 
 
 ... 
 
 ... 
 
 «nn 
 
 «nn+l 
 
 Jf. 
 
 We have accordingl}', in general, from a matrix of the form 
 M, the following relations ; 
 
 Ctrl 
 
 Ou 
 
 «13 
 
 ... Oi, 
 
 ••• «ln ttln+l 
 
 Ot22 
 
 a23 
 
 ••• a^r 
 
 ... a2n «2n+l 
 
 a.2 
 
 a,3 
 
 ... a^ 
 
 ••• am ttm+l 
 
 a«2 
 
 a«3 
 
 ... a„. 
 
 ••• ttnn ttrvn+l 
 
 a,2 
 
 an 
 
 ai3 ••• 
 
 air 
 
 • •• a.|,j ai,,^i 
 
 ^21 
 
 a23 ••• 
 
 a2. 
 
 ••• a2„ a2.+i 
 
 «rl 
 
 a,3 - 
 
 a,. 
 
 "• cim a„,^i 
 
 am 
 
 a«3 ••• 
 
 a„r 
 
 ••• a„„ a«n+i 
 
 + --+(-l)"a^^.i 
 
 ttji (Xj2 
 
 Ctrol (I9Q 
 
 air 
 
 a2r 
 
 am 
 
 a2n 
 
 a_ 
 
 a^i a,jj ... a^ 
 a^i a„2 ••* a„y 
 in which r has successively all values from 1 to n inclusive. 
 
 = 0, 
 
98 THEORY OF DETERMINANTS, 
 
 84. We will now select a few examples to illustrate the 
 foregoing processes from the vast field of application. 
 
 I. To find the condition that three right lines shall pass 
 through the same point. 
 
 Let aiX + 6i2/ + Ci = ■^ 
 
 a2X + ^22/ + ^2 = r (A) 
 
 G'S^ + % + Cg = ) 
 
 be the equations of the lines in cartesian co-ordinates, and let 
 a?!, 2/i be the given point. Equations (A) must be satisfied for 
 x = Xi, y = yi; hence 
 
 cti^i + hVi + Ci 
 
 «2aJi + hVi -f C2 = ^ • (JB) 
 
 (^8^ + ^zjji 4- C3 
 
 0) 
 
 But in that case, by 79, 
 
 Iai&2C3l = 0, 
 which expresses the required condition. 
 
 II. To find the condition that three points shall lie on the 
 same right line. 
 
 Let (a?i, 2/i)» ('»2» 2/2)? (i»8» ^s) 
 
 be the given points, and 
 
 ttjx + 61?/ + Ci = 
 
 the equation of the line. Then 
 
 cfi^^i + hVi + Ci = 0, 
 «fca;2 + %2 + c^ = 0, 
 «ba;3 + 6^3 + 03 = 0. 
 
 Whence the required condition is 
 
 a?i Vi 
 
 1 
 
 x^ 2/2 
 
 1 
 
 ^ 2/3 
 
 1 
 
 = 0. 
 
 («) 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 99 
 
 As an application of the present example, we show that the 
 middle poirits of the three diagonals of a complete quadrilateral 
 lie on the same straight line. 
 
 The three diagonals being 0(7, BA^ ByA^, and their middle 
 points F, D, E, we have to show that F, Z), E are on the same 
 right line. 
 
 Take the vertex as origin, and the sides OAi, OB^ as axes 
 of reference. 
 
 Put a^ = OA, a^^OA,, h^ = OB, h^^^OB^. 
 
 The co-ordinates of D are — , -^, and the co-ordinates of E 
 
 are — , — • The abscissa of F is half the abscissa of (7, and 
 
 2 2 
 
 the ordinate of F is half the ordinate of C. Hence we have 
 to find the co-ordinates of 0. The equations of AB^ and A^ B 
 are respectively 
 
 -+|- = 1, or hcfc ■\- a^y = a^hi\ 
 
 «! 02 
 
 — f- f = 1 , or hx^-\-aoy=^ a^h^- 
 a^ Oi 
 
 Whence the co-ordinates of G are 
 
 
 aib.2 
 
 a. 
 
 
 
 W 
 
 a^ 
 
 x = 
 
 a.A 
 
 as 
 
 ? 
 
 y = 
 
 b, 
 
 &2 
 
 aA 
 ai 
 
 h 
 
 «! 
 
 
 h 
 
 a^ 
 
 
 
 W 
 
 aa 
 
100 
 
 THEORY OF DETERMINANTS. 
 
 and the co-ordinates of F are 
 
 2 ( 62^12 — ^i«i )' 2 (^20^2 — ^lOtl) ' 
 Now, by equation {R) above, 
 
 A = 
 
 «1 
 
 2 
 
 
 
 2 
 
 1 
 
 a, 
 2 
 
 
 
 &2 
 
 2 
 
 1 
 
 aia2(&2- 
 
 ■M 
 
 5i&, 
 
 (02 - «i) 
 
 1 
 
 =0, 
 
 if the three points are on the same straight line. 
 
 A = 
 
 1 
 
 4(62«2 — ^l«l) 
 
 a2 
 
 ttj ttg (62 — <^l) ^1 &2 (<^2 — <^l) ^2 % — ^1 <^1 
 
 Now add the tliird column of this determinant multiplied by 
 — Qi to the fii'St column ; also add the third column multiplied 
 by — hi to the second column. Then 
 
 A = 
 
 1 
 
 4 (62 0^2 — ^itti) 
 
 1 
 
 ag — «! h^ — bi 1 
 
 «! bi («! — ag) tti 5i (6i — 62) ^2 «2 — ^1 «i 
 
 which is obviously zero. Hence F, D, E are on the same right 
 line. 
 
 III. To ohtaiyi the equation of a circle passing through three 
 given points. 
 
 The general equation of the circle is 
 
 {x^ 4-2/^) +2ax +2by 4-c = 0. 
 
 If (a?!, 2/1)? (a^2, 2/2), (a^3. 2/3) are the given points, 
 
 (^1' + 2/1') + 2 aa^i + 2 by, + c = 0, 
 («2' + 2/2') + 2 a.r2 + 2 62/2 + c = 0, 
 W + 2/3') + 2aa;3H- 262/3 + c = 0. 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 101 
 
 These four equations are simultaneous for the parameters 
 
 a, 6, c'f hence, by 79, 
 
 a^ -\-y^ 2x 2y 1 
 
 x^' + y^' 2x, 22/1 1 
 
 Xz^ + V'/ ^^2 22/2 1 
 
 ^i + yi 20^3 22/3 1 
 
 = 0, 
 
 (O 
 
 which is the equation sought. 
 
 That equation C is the required equation of the circle deter- 
 mined by («!, 2/i)? (^2? 2/2)5 (%i 2/3)? is obvious from the form of 
 the first member. The determinant when expanded obviously 
 gives a function of the second degree, and having the charac- 
 teristics which distinguish the equation of the circle. Moreover, 
 this equation is satisfied for x = Xi^ y — 2/1, since in that case 
 the determinant vanishes. The same is true if x^x^^ ^ = 2/2» 
 or a; = a;3, 2/ = 2/3- 
 
 IV. To find the relation connecting the mutual distances of 
 four points on the circle. 
 
 We must have, if the points are (a?i, 2/1) , {x^^ y^}-, {x^-, 2/3) , 
 (^4? 2/4)? ^ determinant equation just like the last one above, 
 except that the first row of the determinant will have the sub- 
 scripts 1, the second row the subscripts 2, and so on, the last 
 row having the subscripts 4. 
 
 Accordingly, multiplying together 
 
 Xi' + Vi' 
 
 -2x, 
 
 -22/1 1 
 
 X 
 
 1 x^ 2/1 «i^ + 2// 
 
 xi^yi 
 
 -2x, 
 
 -22/2 1 
 
 
 1 X2 2/2 a;2" + 2/2^ 
 
 ^i + yi 
 
 -2x, 
 
 -22/3 1 
 
 
 1 X, y, x^^ + yi 
 
 x^^y^ 
 
 — 2x^ 
 
 -22/4 1 
 
 
 1 x^ y^ x^' + y,' 
 
 which are two different forms of the first member of equation 
 {(J) above, we obtain the required relation 
 
 (12)2 ^13)2 (14)2 
 
 (12)2 Q ^23)2 (24)2 
 
 (13)2 (23)2 (34)2 
 
 (14)2 ^24)2 (34)2 
 
 = 0, 
 
102 THEORY OF DETERMINANTS. 
 
 in which 
 {12y = {x,-x,y-\-{y,-y2)\ {13y = (x,-Xsy + (y,-y,y, 
 
 and, in general, {iky is the square of the distance bet^feen 
 the ith and kth. points. 
 
 Expanding this determinant by 63, III., and adding and 
 subtracting 4(12)2(13)2(24)2(34)2, we obtain 
 
 [(12)2 (34)2 _|. (13)2 (24)2 _ (14)2 (23)2]2 
 
 -4(12)2(13)2(24)2(34)2 = 0. 
 
 Whence 
 
 |[(12)(34)-(13)(24)-(14)(23)] 
 
 [(12)(34)-(13)(24)+(14)(23)]^ 
 
 Xl[(12)(34) + (13)(24)-(14)(23)] 
 
 [(12) (34) + (13) (24)-f(14) (23)]J = 0, 
 
 or (12)(34)±(13)(24)±(14)(23) = 0, 
 
 which expresses the condition sought in its simplest form. 
 
 V. To find the condition that two given straight lines in space 
 may intersect. 
 
 (a) Let x — a _ y — /3 _ z — y .jx 
 
 tti ~ bi Ci ' ^ 
 
 x—ai _ y — Pi _ g— yi (2) 
 
 (3^2 ^2 ^2 
 
 be the equations of the lines. If these lines intersect, the 
 
 ^^^^^^ px-\-gy + rz = d 
 
 may be passed through them, and we must have for the first line 
 
 pa +ql3 -{-ry=d\ 
 pai + Q'&i + rci = J ' 
 
 (3) 
 (4) 
 
 and for the second line 
 
 pai 
 
 pa^-\-qb2-\-rc2=0 ^ (6) 
 
 jpai + gft+ryi = (i) (5) &- 
 
\S ■ 
 APPLICATIONS AND SPECIAL FORMS. 103 
 
 From (3) and (5) 
 
 i>(a-ai) + g(;S-ft)+r(7-yi) = 0. (7) 
 (4), (6), (7) being simultaneous, the required condition is 
 
 tti bi Ci 
 
 0,2 V2 C2 
 
 a — tti p — pi y — yi 
 
 = 0. 
 
 (6) If the straight lines are given by the equations 
 aix -{- biy -{- CiZ = di 1 
 
 a2X-\- bzy -^02' 
 
 asX + b.sy-\-CsZ = ds 
 a^x + b^y + C4 
 
 ?3Z = (^3 I 
 
 (1) 
 
 (2) 
 
 these four equations are simultaneous for the point of inter- 
 section (aj, y, 2J) , and the condition of intersection is 
 
 * ^' • 
 
 ' ai62C3<^4l = 0. 
 
 VI. To jind the equation of a plane passing through three 
 given points {x^, 2/1? ^i)^ (^2, 2/25 ^2), (.^3^ Vs^ ^3)- 
 Let the plane be 
 
 aiX-^biy-\-CiZ = di. (1) 
 
 We must have 
 
 'aiXs + biys-hc^Zs=dj_ 
 
 (^) 
 
 Equations (D) and (1) being simultaneous for the para- 
 meters Oi, 61, Ci, di, we have for the equation sought 
 
 X y z 1 
 
 Xi 2/1 ^i 1 
 
 ^2 2/2 ^2 1 
 
 iC3 2/3 % 1 
 
 = 0. 
 
 0/1 Vic 
 
104 
 
 THEOKY OF DETERMINANTS. 
 
 VII. An interesting application of determinants is afforded 
 by the following problems. 
 
 (a) To extend a recurring series of the rtli order without 
 knowing the scale of relation. 
 
 As is well known, a series of the form 
 
 UQ + Uy^x + u^x^-^ h w^.^a;'*-*- H h ^n^"" + — 
 
 is a recurring series if the relation of any r+1 consecutive co- 
 efficients w„, ?f„_i, ••• Un_^ can be expressed by a linear equation 
 (the scale of relation). Under these conditions the series is 
 called a recurring series of the rth order. Every such series is 
 accordingly determined when 2?* of its consecutive terms are 
 known. If all the coefficients, with the exception of the 2 rth, 
 are known, this last is easily found. By the conditions of a 
 recurring series 
 
 ^^r+1 -^PlUr +PlUr-i + P^U^-2 H hPr-1^2 + PAh = 
 
 ^%-l + Pl^%-2+P2W2r-3 + i>3W2r-4H hPr-l^r -^ Pr^r-l^^ 
 
 U2r +i)l^2r-l + i>2^2r-2 + P3^2r-3H b Pr-i^K+l+ Pr'^r =0 , 
 
 Now, by 79, 
 
 (F) 
 
 U^ W^_l W^_2 Wr-3 
 
 Ur^l U^ Ur_i Ur^2 
 
 U2r-\ U2r-2 %r-3 ^2r-4 
 
 ^2r '^2r— 1 '^2r— 2 ^''2r— 3 
 
 U2 Ui 
 
 Us 
 
 lU 
 
 U,.+i u. 
 
 = 0, 
 
 whence Wsr is found by expanding the determinant and solving 
 the equation. 
 
 To find U2r+i we have only to increase each subscript by unity. 
 
 Applying the above process to extend the series 
 
 l-\-x-\-bx^-^nx^ + 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 105 
 
 we find 
 
 
 
 
 
 
 
 5 1 
 
 1 
 
 
 13 5 1 
 
 
 u^ 13 5 
 
 
 13 5 
 
 1 
 
 = 0; 
 
 u, 13 5 
 
 = 0; 
 
 ^5 W4 13 
 
 
 W4 13 
 
 5 
 
 
 Us ^/4 13 
 
 
 Uq Us W4 
 
 whence u^ = Al, W5 = 121, 16,5 = 365. The series is accordingly 
 1 _}_ a; _|- 5ar^ _|- 13a^ + 41 ir* + 121 a;^ + 365a;^ 4- •-. 
 
 (6) To find the generating function for any given recurring 
 series. 
 
 Since a recurring series is always the quotient of two integral 
 functions, of which the divisor is of a degree higher by 1 than 
 the dividend, we may find the required generating function by 
 indeterminate coefficients, as follows : 
 
 Assume the given series 
 
 Wo+ iiriX-\-u^-\ |-t^x= 
 
 (T) 
 
 1 +PlX + P2^-\ ^PrX' 
 
 (after both terms of the fraction have been divided by the first 
 term of the denominator) . 
 
 From the first r of equations (F) of the preceding example 
 we can determine the constants pi, pz-'-Pr- ^e may therefore 
 find the scale of relation. We have from equations {F)y after 
 obvious interchanges of columns, 
 
 Pr = 
 
 — u^ 
 
 Ui " 
 
 ' U^-2 
 
 Ur-1 
 
 - w.+l 
 
 W2 •• 
 
 ' y'r-l 
 
 Ur 
 
 - ?*r+2 
 
 M3 .. 
 
 • U, 
 
 Ur+1 
 
 • 
 
 I 
 
 : 
 
 * 
 
 — W2r-1 
 
 Uf " 
 
 • W2,_3 
 
 W2r-2 
 
 ^0 
 
 Ui" 
 
 • Wr_2 
 
 ^*r-l 
 
 Wi 
 
 U2 " 
 
 • Wr-1 
 
 U^ 
 
 W2 
 
 1h " 
 
 • U^ 
 
 Ur+1 
 
 • 
 
 • , , 
 
 ^ * 
 
 * 
 
 W._l 
 
 U, " 
 
 • W2r-3 
 
 W2r-2 
 
 Having determined the constants PxiPi'-'Pn we need only 
 clear equation {T) of fractions; and then, equating the co- 
 
106 
 
 THEORY OF DETERMINANTS. 
 
 efficients of like powers of x, obtain the usual linear equations 
 from which Oq, aj, Og ... a^_i are found. 
 
 For an example, let us find the generating function of the 
 series we extended in the last example. 
 
 Put 
 
 Here 
 
 Wo=l, Wi=l, U2 = 5, '". 
 
 Substituting in the second member of ( Ti), clearing of fractions, 
 and finding the values of ao and Oj, we find 
 
 f(^)^ 
 
 1 
 
 1— 2cc-3a^ 
 
 85. The coefficients of the quotient Q of two polynomials 
 Pi and P2, and the coefficients of the remainder i?, can always 
 be expressed as determinants in terms of the coefficients of 
 Pi and Pg- 
 
 The method employed in the following example is applicable 
 in general. 
 
 Pj = Oo^ + %^^ + (^2^^ + %^ + <^4^ + % ; 
 P2 = 60^ + ^1 ^^ + ^2^ + &3 ; 
 
 B = VoX^ -\- TiX +r2. 
 
 Let 
 
 Now 
 hence 
 
 P,Q-\-B = Pi; 
 
 (i>) 
 
 O'A- &2f?l + &l52 + ^l, 
 
 a«= Z'89'2 + ^2* 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 107 
 
 From the first three of equations {p) we can find Qq, gi, ^2? and 
 then taking the first three with each of the others in succession, 
 we obtain Tq, r^, i^. For example, 
 
 &o'92 = 
 
 h 
 
 
 
 ao 
 
 h 
 
 &o 
 
 (h 
 
 h 
 
 h 
 
 ^2 
 
 ; Wn= 
 
 h 
 
 
 
 
 
 tto 
 
 h 
 
 ^>o 
 
 
 
 «'i 
 
 h 
 
 hr 
 
 bo 
 
 a2 
 
 h 
 
 h 
 
 h 
 
 ^3 
 
 Let the student find the remaining coeflScients. 
 
 86. The coefficients of any equation can be expressed in 
 terms of the roots as the quotient of two determinants, as fol- 
 lows. The method employed is applicable in general. By 
 reference to examples 6 and 7, page 37, it is readily seen that if 
 
 f{x) = ic^ — a^T? + a2X — ttg = {x — a){x — (3) (x — y)^ 
 
 we have 
 
 0' / 
 
 = - (^-y) (7-«) («-|8) (x-a) {X-P) (x-y) . 
 
 Expanding the first member, 
 
 P y 
 
 +x 
 
 1 1 
 
 a P 
 
 1 1 
 1 
 
 y 
 
 /3^ / 
 
 1 
 
 •a^l 1 
 
 1 1 
 
 + a^ 
 
 1 1 1 
 
 ^ r 
 
 
 a /3 y 
 
 ^ / 
 
 
 a^ /5^ y 
 
 {a^~aiX^ -{-a2X — as), 
 
 From this identity the required expressions in determinant form 
 are at once obtained by equating the coefficients of like powers 
 of X. 
 
 87. With the aid of determinants we readily find the sum of 
 the like powers of the roots of any equation, as follows : 
 
108 
 
 THEORY OP DETERMINANTS. 
 
 Let Si, S2, S3...s„ denote as usual the sum of the first, sec- 
 ond, ... nth powers of the roots of 
 
 Then from the theory of equations we have 
 
 0. 
 
 (1) 
 
 Pi +Si 
 
 2^2 +P1S1 + S2 
 ^Pa +P2S1 + i>i«2 + «3 
 
 = 
 = 
 = 
 
 {n-l)Pn-l + Pn-2Sl + Pn-sS2-\-Pn-iS3-\- "' + 5^-1 = 
 
 nPn +Pn-lSl + Pn-2S2 + Pn^3S3-\ ^ PlSn-l+ S,,= 
 
 From equations (S) we obtain at once 
 
 {S) 
 
 Pi 
 
 1 
 
 . 
 
 ..0 
 
 2P2 
 
 Pi 
 
 1 . 
 
 -00 
 
 3i>3 
 
 P2 
 
 Pi ' 
 
 ..0 
 
 (w-l)Pn-l Pn-2 Pn-S '"Pi 1 
 nPn Pn-1 Pn-2 '" P2 Pi 
 
 If in (1) the coefficient of of had been p^y we should, of course, 
 
 have to write in the value of s„ just obtained, {^^^^—\ instead 
 
 \Pj 
 of ( — 1)", and pQ instead of 1, for each element of the minor 
 
 diagonal of the determinant. If n = 3, and n= 4, the above 
 
 formula gives 
 
 So= — 
 
 Pi 1 
 2i)2 Pi 1 
 
 3i?3 P2 Pi 
 
 , and 54 = 
 
 respectively. 
 
 Pi 1 
 
 2p, p, 1 
 
 3i>3 P2 Pi 1 
 
 ^Pa Ps P2 Pi 
 
 88. Equations (S) can also be employed to give the value 
 of the coefficients in terms of s„ Sg? ^3... ^n? ^7 solving these 
 equations for the coefficients. We find 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 109 
 
 JP. 
 
 _(-iV 
 
 Si 
 
 1 
 
 
 
 . 
 
 .. 
 
 
 
 S2 
 
 Si 
 
 2 
 
 . 
 
 .. 
 
 
 
 Ss 
 
 S2 
 
 Si 
 
 3 . 
 
 .. 
 
 
 
 ^n-3 •'n-4 
 
 n-1 
 
 Sl 
 
 If, as before, the coeflScient of a;** in equation (1) had been Pq, 
 we would write in this value ofp^, ( ^^^ ) instead of (^^— ] • 
 If n = 3, and 7i = 4, 
 
 1 
 
 i>3 = -^ 
 
 Sl 1 
 
 
 
 52 Sl 
 
 1 
 
 S3 S2 
 
 Sl 
 
 ; i?4= 
 
 Sl 
 
 1 
 
 
 
 
 
 S2 
 
 Sl 
 
 2 
 
 
 
 S3 
 
 S2 
 
 Sl 
 
 3 
 
 Si 
 
 S3 
 
 S2 
 
 Sl 
 
 89. Any differential equation of the form 
 
 2/3 -h X12/2 +^22/1 +Xsy = 0, 
 
 (1) 
 
 in which y, 2/1? y^i Vz denote a function of x and its successive 
 derivatives respectively, and Xi, Xg, X3 are also functions of ic, 
 can be reduced to an equation of the next lower order, provided 
 a particular solution of (1) is known. 
 Let y = z satisfy equation (1). Then 
 
 Put 
 
 z^ + X^z.,^-X2Z^-\-X^z = 0. 
 
 u = y^-'-y, v = zu. 
 
 (2) 
 
 Then, as above, denoting derivatives by subscripts, we have 
 
 — V -\-zyi-Ziy = 0. 
 
 — Vi-^zy2-Z2y=0. 
 
 — '^2 + ^ys + Ziy2 — 2=22/1 — 23?/ = 0. 
 
 These three equations and (1) are simultaneous ; hence 
 
110 
 
 THEOBY OF DETERMINANTS. 
 
 A= 1 Xi X, X,y =0. 
 
 z —V —z^y 
 
 2; —Vi—Z2y 
 
 z zi -Z2 —Vz-z^y 
 
 Now multiply the fourth column of A by - , then add to the 
 
 fourth column the first multiplied by z^, the second multiplied 
 by 22, and the third multiplied by ^i, and we have 
 
 = 0; 
 
 or V2Z + Vi{Zi—XiZ)-\-v{z2 + X2z) = 0, 
 
 which is a differential equation of the second order. 
 
 1 
 
 X, 
 
 X2 
 
 
 
 
 
 
 
 z 
 
 — V 
 
 
 
 z 
 
 
 
 -Vi 
 
 z 
 
 «i 
 
 -Z2 
 
 -V2 
 
 Resultants, or Eliminants. 
 
 90. If we have given a system of n homogeneous equations 
 containing n variables, or, what amounts to the same thing, 
 n non-homogeneous equations containing n — 1 variables, it is 
 always possible to combine these equations in such a way as to 
 eliminate the variables and obtain an equation of relation be- 
 tween the coefficients of the form 
 
 E = 0. 
 
 (1) 
 
 i?, when expressed in a rational integral form, is called the 
 Resultant or Elmiinant of the system. In 77 and 79 we 
 l)ointed out the fact that the equation i? = must hold be- 
 tween the coefficients of a system of equations if they are 
 consistent with each other (simultaneous). In the examples 
 of 84 we repeatedly found the resultant of given systems of 
 equations. Among the most important problems of elimination 
 is the following : to find the resultant of two given equations, 
 containing a single variable. 
 
,A 
 
 APPLICATIONS AND SPECIAL FORMS. Ill 
 
 We consider first 
 
 Euler's Method of Elimination. 
 91. I. Given 
 
 f{x) =Pq3? -f- p^x 4-P2 = 0, (1) 
 
 and <^{x) = q^x^ -^ q^x + q^. (2) 
 
 If these equations have a common root, we must have 
 
 f{x) 4>{x) 
 
 ^— ^ = (a^x + as) , ^3^. = (61a; + 62) , 
 
 in which %, ag, 61, 62 are undetermined, since r is unknown. 
 Then 
 
 {\x + 62) {p<iX^-\-PiX +P2) = (^la; + tta) (go^+ gi^; + ga)- 
 
 Whence the equations 
 
 \Pq+ -aigo+ =0. 
 ^iPi + 2>2i>o - tti^i - a2g'o = 0. 
 
 &li>2 + &2i3l — «ig2 — «2gi = 0. 
 
 +&2i>2+ — a2g2 = 0. 
 
 Hence, by 77, the resultant is 
 
 jR 
 
 = 0. 
 
 Po go 
 
 i>i i>o gi go 
 P2 Pi g2 gi 
 P2 g2 
 
 II. In general, let 
 
 f{x)=p,x- +p^x--' +p,x^-^ + ... +Pr,_^x+p^=0. (1) 
 <^(a;) = q,x- + gia;'*-' + g2a;'*-' + - + qn-i^ + g« = 0. (2) 
 
 Let r be a common root of (1) and (2), and put 
 
 —-^a,af^-^-\.a,x--'+>.>^a^_^x + a^=f,{x), 
 |^= 6,a--i + &2CC-2 + ... +\_^x+K^<i>,{x), 
 
112 
 
 THEORY OF DETERMINANTS. 
 
 Id which the coefficients Oi, dg? •••««? ^i? &2) "' K ^-re unde- 
 termined. Then 
 
 f,(x)cl>(x) = <l>,{x)f{x). (I.) 
 
 From the identity (I.), by the theory of indeterminate co- 
 efficients, we must have m + n homogeneous equations between 
 the m + n coefficients ctj, ag'-'C*^, bi, 62 •••6„. Hence the de- 
 terminant of the system of these m-{-n equations must vanish 
 if (1) and (2) have a common root, and the resultant sought is 
 accordingly this determinant. 
 
 As an application of Euler's method, take the following 
 example. To find the conditions that must be fulfilled when 
 
 f{x)=poX^+PiX^-hP2X+Ps=0, (1) 
 
 <f>(x) = qoi)(^-\-qiOi^-{-q2X + qs=0, (2) 
 
 have two common roots. 
 
 If (1) and (2) have two common roots, two factors off{x) 
 must be the same as two factors of <}){x). Hence 
 
 (ax+b) {po^ -^Pi^ +P2^ +P3) = (ca; + d) {q^ + qi^-^-q^-^q^) , 
 where a, b, c, d are indeterminate coefficients. Whence 
 
 apo-\- —cqo+ =0. 
 «i>i + bjio — cqi — dgo = 0. 
 opa + bpi — cq.2 — dqi = 0. 
 «i>3 + bP2 — CQs — dq2 = 0. 
 +bps-{- -^^3 = 0. 
 
 From every four of these five homogeneous equations we obtain 
 a determinant of the fourth order whose vanishing expresses 
 one of the required conditions. Hence the conditions sought 
 are expressed by the matrical equation 
 
 Po Pi P2 Ps 
 
 Po P, P2 Ps 
 
 Qo Qi <?2 ga 
 
 qo qi ^2 Qs 
 
 = 0. 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 113 
 
 Sylvester's Dialytic Method of Elimination, 
 92. I. Given 
 
 PO^ -^Pl^-\-P2^ + Pi = 0, 
 
 qoX^ + qiX +^2 =0. 
 
 (1) 
 (2) 
 
 Multiply (1) successively by a;^, ic, and (2) by a^, x^^ x. Then 
 we have the following system of equations : 
 
 PoX^+PiX^+PoX^+PsOi^ =0. 
 
 Pq^ +PiX^ +P2^-\-Ps^ = 0. 
 
 qoO^ + qj_x^ -}- q^x^ =0. 
 
 qo^:* -\- qiO(^ + q20!^ =0. 
 
 qo^ + qiX^-\-q2X = 0. 
 
 We may consider these equations linear and homogeneous with 
 respect to x^, «*, a^, a^, x, considered as separate variables. 
 Hence 
 
 E 
 
 PO Pi P2 Ps 
 
 Po Pi Pi Ps 
 
 go gi g2 
 
 go qi q2 
 
 go gi g2 
 
 = 0. 
 
 II. In general, let 
 
 f{x)^poxr-j-p,x^-''-{- 
 
 cf>{x) = qoX'' 4-gia;"-i + 
 
 -hqn-1^ + g» = o. 
 
 (1) 
 
 (2) 
 
 If we multiply (1) successively by ic, x^'"X'^, and (2) succes- 
 sively by ic, a^-'-ic"*, we obtain a system of m-j-n equations, 
 linear and homogeneous, with respect to a;, x^, x^^ ...a;'«+« con- 
 sidered as separate variables. From these equations we elimi- 
 nate the variables by 77 and obtain the resultant in the form of 
 a determinant of order m-\-n. 
 
114 
 
 THEORY OF DETERMINANTS. 
 
 B= Po Pi P2 '" Pn Pn+l Pn+2 '" = 0. 
 
 Po Pi '" Pn-l Pn Pn+l 
 
 Po '" Pn-2 Pn-1 Pn 
 
 Qo qi 92 '" Qn 
 
 Qo Ql ••• Qn-l qn 
 
 go ••• qn-2 g„-i qn 
 
 It is evident from the form of R that the coefficients of (1) 
 enter R in the degree of (2), and that the coefficients of (2) 
 enter R in the degree of (1). 
 
 Cauchy^s Modification of Bezoufs Method of Elimination. 
 93. I. Given 
 
 Po^'^+Pl^+P2X+P3=0, (1) 
 
 and qoX^ + qi^^ + ^2^*^ + ^s = 0. (2) 
 
 Transposing and dividing (1) by (2), we obtain successively 
 
 PO _ Pl^+P2^+P8 
 
 go qi^ -{- q2x -h qs 
 
 PqX +i>l ^ P2^-hPn 
 
 qox-\-qi ~q2^ + q^ 
 
 Pq^+PiX-^P2 ^ PS^ 
 
 qo^-hqi^ + q2 qi 
 
 Clearing these equations of fractions, we have 
 
 (Po^i - q^Pi)^ + (i?og2 - go2>2)« + {Poq^ - qoPs) =o, 
 
 iPoq2-qoP2)^-^ liPoq-s-qoPs) + (2>ig2-giP2)]a;+ {Piqs-qiPs)^^, 
 {Poq-s - qoPs)^^ + {Piqs - qiPs)^ + (^2^3 - ^2^3) =0. 
 
 Eliminating o^ and cc, regarded as distinct variables, from 
 these equations by 79, we find 
 
6- 
 
 APPLICATIONS AND SPECIAL FOKMS. 
 
 115 
 
 R = 
 
 \P(^qi\ li>o^2l Ipogsl 
 
 1^0^21 \Poqi\ + \piq2\ li>i?3l 
 
 IPogsl li>1^3l 11^2^3! 
 
 = 0. 
 
 The resultant is found by this method in the form of an axisym- 
 metric determinant,* whose elements are easily written, as we 
 shall show by another example. Let the given equations be 
 
 Po^^+Pix 
 and go^'* + 5'i^ 
 
 '+P2 
 ' + q2 
 
 x'-\-P3X-hPi=0, 
 
 x^ + q^x-\-q^ = 0. 
 
 
 (1) 
 
 (2) 
 
 We have, as before, 
 
 
 
 
 
 ^0 
 
 
 q^x^ -\- q.;^X' + q.^a 
 
 
 
 q^x + qi 
 P(^^-\-PiX-^P2 
 
 
 P2^-\-p^x+p^ 
 qzx" -\- q^x -\- q^ 
 
 PsX-\-Pi 
 qsx + q^ 
 
 ' 
 
 -• (E) 
 
 2JqX^ + p^a^ + p^x 
 q^a^ -{- q^a^ -\- q^x 
 
 + ^3 
 
 _Pa 
 
 qi 
 
 
 
 Clearing equations (E) of fractions, we have 
 
 \poqi\x'^ + \Poq2\x^ + \Poq3\x + \poq4\ 
 
 0, 
 
 \poq2\x^ + UPoQi\-^\PiQ2\^x^ + UPoqA\ + [Piq3\lx-{-\piq^\==o, 
 
 ll>0 53laJ'^4-[li>0^4l + li>1^3l]a^+[li>ig2H-IP2^3l]^+li>2g4l=0, 
 
 \Poq4\^-{-\piqi\x^-\-\p2qi\x-{-\psq4\ =0. 
 
 Hence, as before, the resultant is 
 
 E = 
 
 \po qi\ 
 
 \Poq2\ 
 
 \Poqs\ 
 
 \Poq2\ 
 
 \Poq3\ + \piq2\ 
 
 \poqi\ + \piqs\ 
 
 IPo ^sl 
 
 Ipoq^l-hlpiqsl 
 
 \p1qi\-h\P2q3\ 
 
 Ipo q4\ 
 
 \pi qi\ 
 
 \P2 qi\ 
 
 \poqi\ 
 
 \pi qi\ 
 \p2qi\ 
 IPs q^l 
 
 = 0. 
 
 For symmetrical determinants, see 107. 
 
116 
 
 THEORY OF DETERMINANTS. 
 
 To form this resultant directly from the equations, write the 
 ro symmetrical determinants 
 
 
 \Poqi\ \poq2\ \Poqs\ \poqi\ 
 \poq2\ Ipoq^l \poqi\ Ipiq^l 
 \Poq&\ Ipoq^ lihq^l \p2qi\ 
 ipoq^l Ipiq^l lp2q^ Ipsqil 
 
 , and 
 
 \Pi q2\ \Pi qsl 
 \Pi q^l \P2 qs\ 
 
 f 
 
 formed from the coefficients of (1) and (2) in an obvious and 
 easy way. It is then evident that B is formed from these two 
 determinants by adding the elements of the second to the four 
 inner elements of the first. If the equations are of the fifth 
 degree, the student will form the resultant in the same way 
 from the three determinants 
 
 iPoqil \Poq2\ \Poq3\ \i>oq^\ Ipoqsl 
 
 iPo^J IjPogsI Ipoq^l Ipoqsl li^i^sl 
 
 iPo^sl \Poq^\ IPogsI IPl^sl 11^2^51 
 
 Ii>og4l Ipogsl \piq5\ l^gsl Ip^qsl 
 
 \Poqo\ Ipiqsl Ipzqsl \psq5\ Ip^qsl 
 
 \piq2\ Ipiqsl \piqi\ 
 \piq3\ \piqi\ \p2q4\ 
 \piqi\ \p2q4\ Ipzqil 
 
 \p2q3\y 
 
 by adding the third to the middle element of the second, and 
 then adding the elements of the second to the nine inner ele- 
 ments of the first. This process is, of course, general. 
 
 From the preceding examples we see that by Bezout's method, 
 tJie resultant of two equations, each of the nth degree, is a sym- 
 metrical determinant of the same degree whose elements are either 
 determinants of the second order or the sum of such determinants. 
 
 II. If the two equations are not of the same degree, suppose 
 we have given 
 
 i>oa^*+Pi^^+i>2^+i>3a;+P4 = 0, (1) 
 
 qox'^-hq^x +^2 =0. (2) 
 
 Multiply (2) by x^ ; the equations are then 
 
 Po^+Pi^^+Pzx'-hPsX-hPi^Oy (li) 
 
 qox'-\-q,x'-\-q,x' =0. (2,) 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 117 
 
 From (li) and (2i), 
 
 PO^ +Pl ^ P2^'^ + P3^ + j^4 ^ 
 
 Clearing these equations of fractions, we have 
 
 \po q2\^ + \ \pi ^2' - Q'oi>3i ^ - (^oi>4 + qiPs)x - qip^ = 0. 
 
 With these equations consider (2) multiplied by x, and (2), 
 
 qQx'^ + qiX^-^q2X = 0, 
 qooc^-^qix + q.^ =0. 
 
 From these four equations eliminate x^, o^, cc, and we have 
 
 B 
 
 1^0^21 
 
 
 \poq2\ qoPs qoPi 
 
 Piq2\-qoP3 qoPi + qiPs qiPi 
 qi -q2 
 
 qo 
 
 •qi 
 
 -q' 
 
 0. 
 
 III. In general, let 
 f(x) =poQcr +piaf-i +p2a;«-2 4. 
 
 + Pm~lOC+Pm=0, (1) 
 
 <f^(xy= q,x^ + gio;'-^ + q,^-^ + - + ^n-i a; + g, = 0, (2) 
 
 in which m is greater than n. Multiply (2) by mf'^ ; then 
 (2) becomes 
 
 goa;'" + gia;«-i + g2aj'""^+ ••• + gH-i^'"*"'*^^ + gna;"*""- (2i) 
 From (1) and (2i), 
 
 go qix""-^ + q2^ "" + ••• + gn-i a;"* "-^^ + g„a;- «' 
 
 go^j + gi g2aj"-' -{- gga^"' + • • • + qn-x^ + gn«"~"' 
 
 Poa?"~^+;>ia;"-^+ ... 4.p^_2aj_|_p^_^ p^a;'"-"-f-p„+iaj™-"-i+ ... 4-2?^ 
 
 go«'*"''+gi»"''+ - +gn-2aj+g„-i 
 
 g„a;- 
 
118 THEORY OF DETERMINANTS. 
 
 Clear these equations of fractions, and consider with them 
 the following m — n equations obtained from (2) bj' multiply- 
 ing it in order by 1, «, a^, •••a;"'""""^, 
 
 qoOf-'^ + qiixr '- + q2X"'-^-\ hgn-iic"*~"+ ^n^^'""^ =0, 
 
 go«" +Q'i«"~^H \-qn-i^ +qn =o. 
 
 From these m equations the resultant is obtained by elimina- 
 ting the m — 1 successive powers of x regarded as separate 
 variables. 
 
 The Resultant in Terms of the Roots, 
 
 94. Given 
 
 / = j9oa;"'-FPia;"-^H f-i?«-ia;+i)« = 0, (a) 
 
 <^=go»"+gia;"-'4- - +qn-i^-\-qn =0. (&) 
 
 If tti, a2, ...a^ are the roots of (a), and /3i, ^2-, ...^n are the 
 roots of (6) , we have, of course, 
 
 <t> = qo{x-(3,){x-p2)'"{^-^n)- (h) 
 
 Now, if in go«''* + Q'i^'*~^H hO'n-i^ + Q'n we substitute 
 
 successively ai, og, ... a^, </> takes the m corresponding values, 
 <^(ai), </)(a2) ...<^(a^). With these m values as roots we can 
 form an equation of the mth degree in <^. This equation may 
 be found as follows. Forming the resultant of 
 
 PoX'^+PiX'^-'^-^ \-Pmi^+p„, =0, (1) 
 
 qoX^'+qiX--^ +"'+qn-iX+qn-<l>=-0, (2) 
 
 bv 92, we have 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 119 
 
 B,= 
 
 PO Pi P2"' Pn Pn+l Pn+2 
 
 Po Pi--' Pn-1 Pn Pn+l 
 
 i>o ••• Pn-2 Pn~l Pn 
 
 Qo ^1 Q2 ••• 9n-<f> 
 
 go Qi ••• Qn-i qn-4> 
 
 ^0 — 5n-2 g«-i qn-4> 
 
 = 0. 
 
 This is obviously an equation of the mth degree in <^, whose 
 roots are <^(ai), <^(a2), c^Cag), ••• </>(a,„). The absolute term T 
 of this equation is the product of its m roots multiplied by a 
 factor. 
 
 But from the determinant i?i, 
 
 Again, since JKi becomes identical with { — lyR of 92, II., 
 when we have made <^ vanish, we see that 
 
 In just the same way we can show that 
 
 2"= (-i)>V(/80/(ft) -/(/S.); 
 
 and hence, after suitable interchanges of lines, 
 
 95. These forms of the resultant R may be obtained by 
 symmetric functions, as follows : 
 
 /W=i>oa;" + Pia;'^-'+i?2a^""'+ ••• +Pm-iX+p,,=:0, (a) 
 <i>{x) = q,x- + q^x--^ + q^x^-'' + ... + q,,^.,x + q,, = 0. (6) 
 
 Then aj, ag, •••a^ being the roots of (a), and ft, ft, ."ft 
 the roots of (&), 
 
120 THEORY OF DETERMINANTS. 
 
 Now, if (a) and (6) have a common root, the product 
 
 /(A)/(/SO-/(A) = .P 
 
 must vanish, since in that case some one of the factors vanishes. 
 The same statement applies to the product 
 
 c/>(ai)<^(a2)...(^(aJ=Pi. 
 
 But /(ft) = po (ft - ai) (ft - a^) . . . (ft - a J , 
 
 /(ft) =Po (ft - ai) (ft - a,) ... (ft - a„) , 
 
 /(ft) =Po(ft - ai) (ft - a^) ... (ft- aj ; 
 
 also </) (ai) = qo (ai - ft) (ai - ft) . . • (aj - ^„) , 
 
 </> (02) = go («^ - ft) {0.2 - ft) — (a2 - ft,) , 
 
 <^(am) = go(a.-ft)(a.-ft)-(an.-ft.). 
 
 P is accordingly made up of mn factors of the form ft — a,. 
 We may therefore write 
 
 P=jOo"n(ft-a,), 
 
 where r has all integral values from 1 to w, and s has all 
 integral values from 1 to m. P is moreover a symmetric func- 
 tion of the roots of <^ (a;) =0, and can therefore always be 
 expressed as a rational integral function of the coefficients ; 
 and since it vanishes when /(a;) = and cf){x) = have a 
 common root, and not otherwise, when P is expressed in terms 
 of the coefficients, P is the resultant of (a) and (6) . In the 
 same way 
 
 Pi = go'"n(a,-ft) = (-l)-«go"n(ft-a.), 
 
 where s and r have the same values as before. Hence we may 
 write the resultant 
 
 J? = (-l)-go'"/(ft)/(ft)-/(ft.)=l>o''<^(ai)c^(a2)...<^(aJ, (A) 
 
APPLICATIONS AND SPECIAL FORMS. 121 
 
 for both these expressions are rational integral functions of the 
 coefficients of f{x) and <^(a^), which vanish when /(a;)=0 
 and <^(a;) = have a common root, and not otherwise, and 
 -wrhich become identical when expressed in terms of the co- 
 efficients. The value of R can accordingly be written 
 
 Properties of the Resultant. 
 
 96. I. By reference to the forms {A) , we observe that the 
 coefficients po? lh'"Pm of equation (a) enter the resultant in 
 the 7ith degree, and the coefficients q^^ Q'i-"5'» of (6) enter the 
 resultant in the mth degree ; moreover, we readily see that 
 (_l)""*gy'»j>^'* is a term from the first form of the resultant, and 
 Pq- q,^ is a term from the second form ; hence, given two equa- 
 tions of degree m and n respectively^ the order of the resultant R 
 in the coefficients is m-\-n; the coefficients of the first eqiiation 
 are found in R in the degree of the second, and the coefficients of 
 the second equation enter R in the degree of the first. 
 
 II. If the roots of (a) and (6) are multiplied by A;, R is 
 multiplied by A;'"". Since each of the mn binomial factors of 
 
 is in this case multiplied by A:, the truth of the statement is 
 obvious. This result is frequentlj' expressed by saying the 
 weight of the resultaiit is mn.* 
 
 III. If the roots of (a) and (b) are increased by 7i, the resul- 
 tant of the transformed equations is the same as the resultant of 
 the original equations. This, too, is obvious, for none of the 
 factors of R is changed when both roots are increased or 
 diminished by the same number. 
 
 * By the weight of any term is meant the degree in all the quantities 
 that enter it. The weight of ab^c^ is 6. 
 
122 THEORY OF DETERMINANTS. 
 
 IV. If the roots of (a) and (b) are changed into their recip- 
 rocals, the resultant Ei of the transformed equation is ( — lj'"'*i?. 
 
 Putting y = —, (a) and (6) become respectively 
 
 X 
 </>(2/) = Qny"" + Qn-l 2/""' + Qn-2 2/""' + " ' + QiP + Qo == 0. {b,) 
 
 Whence 
 
 i2x = gri>."n(i-i) 
 
 But 
 
 (aia2---a„)= , (/Ji/:^2 '•• Pn) = » 
 
 hence the resultant of the transformed equations is identical with 
 the resultayit of the original equations^ or differs from it only in 
 , sign, according as mn is even or odd. 
 
 97. Of all the methods of elimination given, the dialytic 
 method is the most direct. Another advantage of this method 
 is that it may obviously be employed to eliminate one of two 
 unknowns from a pair of equations, as in the following example. 
 
 Given 
 
 Po^-{-Pi ^^y +P2^y^ +Psf = 0, 
 
 qo^ + qixy ^q.f +93^ =0. 
 
 / 
 
 To eliminate x we form the following equations : 
 
 Pox!* -hPi^^^y ■i-P2^y^ +P3xy^ = 0, 
 
 Po^ -^Pi^y -hP2xy--hP3f =0, 
 
 qox*-\-qix^y-i-{q2y' -\-<i3)x^ =o, 
 
 QoO^ +qiXy + q.y--\-q.;^=^0. 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 123 
 
 Whence 
 
 Po 
 
 PiV 
 
 i>2/ 
 
 Psf 
 
 
 
 
 
 Po 
 
 PiV 
 
 P2y^ 
 
 Psf 
 
 5o 
 
 qiv 
 
 q2f-\-q3 
 
 
 
 
 
 
 
 Qo 
 
 QiV 
 
 g2y'-\-q3 
 
 
 
 
 
 
 
 go 
 
 qiv 
 
 g2y' + qs 
 
 0, 
 
 an equation containing only y» 
 
 98. The same method is also frequently applicable to the 
 elimination of w — 1 unknowns from a set of n equations, so as 
 to obtain a final equation with but one unknown. It will afford 
 the student a good exercise to find from the three equations 
 
 aia^y-\-a2Xz-^as =0, (1) 
 
 yz — a^x = 0, (2) 
 
 a^xy +aQX +a7 =0, (3) 
 
 a final equation in y, as follows : First, eliminate x from (1) and 
 (3), and also from (1) and (2), obtaining two new equations 
 in y and z. From these equations eliminate », and obtain 
 
 a^y'^-^a.^a^ a^a^ 
 
 —{a^+ae)a2a7 aia^'^+{a^aT^-{-2a^a5aQ)y+a^ai 
 
 an equation in y of the sixth degree. ~ ' 
 
 99. A further interesting application is found in the follow- 
 ing examples, in which three variables are eliminated from as 
 many equations. Given 
 
 a^i + 3^2 + ^3 = 0, x^ = a^ x}—h^ x^ = c. 
 
 Multiplying the first equation successively by 
 
 a?!, x^-) Xq^ a;ia;2iC3, 
 
 and substituting from the last three, we get 
 
 CL ~\~ x^ X2 ~j~ a?! x^ ^^ u , 
 
 b+XiX^ -f- a!2a;3 = 0, 
 
 C "7~ 37 J ajg -j~ X2 X^ ^= ' J , 
 
 cxi X2 -\- hxi x.^ -\-ax.,x.^z=0. '' 
 
124 
 
 THEOKY OF DETERMINANTS. 
 
 Eliminating XiX^y XiX^^ ajg^a? 
 
 a 1 
 
 1 
 
 
 
 b 1 
 
 
 
 1 
 
 c 
 
 1 
 
 1 
 
 c 
 
 b 
 
 a 
 
 = 0, 
 
 Had we multiplied the first equation successively by 
 
 i^ iC2*^3? *^1*^3? ^l«^2? 
 
 we should find by eliminating XiX2X^, Xi, x^^ x^^ 
 
 0. 
 
 
 
 1 
 
 1 1 
 
 1 
 
 
 
 c 6 
 
 1 
 
 c 
 
 a 
 
 1 
 
 b 
 
 a 
 
 If the original equations are 
 
 Xi-\-X2-\-X^=0^ Xi^=a, X2^=b, 073^ = c, 
 
 one form of the resultant is obtained by multiplying the first 
 equation successively by 
 
 Xi^ ^2? "^S? *^2 ^3 » •^1*^3? "^i "^2 ? "^1 •^2'''3? 3/j*^2 "^S? 2/13/23/3 j 
 
 and substituting from the last three. Then by eliminating 
 
 •'^l ) *^2 ) 3/3 J flJgflJs) fl72 3?3} 3/1 37^^ X^X^ X^ ^ X^ X^X^ j 37j SJg flJg^ 
 
 we find 
 
 10 110 
 10 10 10 
 1110 
 
 
 
 c 
 
 6 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 c 
 
 
 
 a 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 6 
 
 a 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 a 
 
 
 
 
 
 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 b 
 
 
 
 1 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 c 
 
 1 
 
 1 
 
 
 
 = 0. 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 125 
 
 100. For a final application of the dialytic method we select 
 the following. 
 
 Given Vc/oic + <^^i + V6o^ + ^1 + ^0 = 0, 
 
 to free the equation from radicals, we ma\' proceed as follows. 
 Put Vao» + ai = 2/i5 V&oa;-f 61 = 2/2. 
 
 Then we get at once 
 
 2/1+2/2 +c=0, 
 
 2/1^ — Ooa; — ai = 0, 
 2/2^ — 60 a; — &i = 0. 
 
 From (1) and (3), 
 
 1 —boX — bi 
 1 2/1 + Co 
 
 1 yi + Co 
 
 = 0. 
 
 (1) 
 
 (2) 
 (3) 
 
 (4) 
 
 = 0, 
 
 Eliminating 2/1 from (2) and (4) , we have 
 
 1 2cq Cq —\x—hi 
 
 1 2 Co CQ—h^x — hi 
 
 1 — ofo — % 
 
 1 —aii!— tti 
 
 which is the equation sought. 
 In general, given 
 
 PiVfx +p,Vf(x) -hPs'^W) + - -^pJVfM = B, , 
 
 in which ri, Vz-'-r^ are integers, and fi{x), f{x) --fnix) are 
 rational integral functions of x, we may rationalize the expres- 
 sion as follows. Put 
 
 /l(^) = 2//S f2{^) = 2/2^"', - /n(a^) = Vn 
 
126 
 
 THEORY OF DETERMINANTS. 
 
 Then we have a system of n equations, from which, together 
 with 
 
 we eliminate the n variables 2/1, Vit ••.2/»» ^^^ obtain a resulting 
 equation in x without radicals. 
 
 Discriminant of an Equation, 
 101. I. Given 
 
 f{x) =p,x-+p,x--''-\-p^x^'-^+ ... +i>„_ia;4-i>n = 0, (1) 
 and the first derivatives of f{x) , or 
 f{x) = np^--^-\- {n-l)p,x^-^+ (^_2)p2»--3+...4.p^_,. (2) 
 
 Then the resultant R of f{x) = and f\x) = is called the 
 discriminant of f(x) = 0, since, if B vanishes, f(x) = and 
 /'(if) = have a common root, and hence f(x) = has equal 
 roots. 
 
 Forming the resultant of (1) and (2) by 92, we have 
 
 Pn-2 Pn-1 Pn . 
 
 Pn-S Pn~2 Pn-l i^n • 
 Pn-i Pn-3 Pn-2 Pn~l Pn . 
 
 2Pn-2 JPn-1 0. 
 
 ^Pn-S ^Pn-2 Pn-1 0- 
 
 4i>„-4 ^Pn-i 2p„_2 p„_i . 
 
 I 
 
 in which the first {n — 1) rows are formed from the coefficients 
 of (1), and the last n rows from the coefficients of (2). 
 
 Now multiply the first row of B by ?i, and subtract it from 
 the nth row ; the 7ith row becomes 
 
 Po 
 
 
 Pi 
 
 
 P2 
 
 
 
 
 Po 
 
 
 Pi 
 
 
 
 
 
 
 
 Po 
 
 npo 
 
 {n 
 
 -^)P1 
 
 (n 
 
 -2)i)2 
 
 
 
 
 npo 
 
 (n 
 
 -l)i>l 
 
 
 
 
 
 
 
 nPo 
 
 
 
 j>i -2i>2 (^-2)i)„_2 -(n-l)i?„_i -np^ 
 
APPLICATIONS AND SPECIAL FORMS. 12T 
 
 Hence R is at once reducible to a determinant of order 2n — 2 
 multiplied \)y Pq] calling this determinant A, we have 
 
 / 
 Now i2 =i>o'*-V'(aO/'(a2)/'(a3), -/'(«„) ; (94 or 95, A) 
 
 , ^ f(x) nx\^yf(x) f(x) 
 
 and since f\x)=--^-^-^ +^-^-^+^-^~L j^ ... ^ :^\±, 
 
 /'(ai) =i>o(ai — «2)(ai — as) ••• («! — »„ -i) (cti — a^) 
 /'(as). =i>o(a2 — ai)(a2~a3) * ' ' («2 — «n-i) ("2 — aj 
 
 f{o-n-\) =i^o(an-i — ai) (a„_i — as) ••• (a„_i — a^.g) (a„_i — aj 
 /'(a„) =Po(an — ai)(an— eta) ••' (««— an-2) (a» — a„_i) 
 
 {E) 
 
 If we multiply equations {E) together, we see that the second 
 member of the result will contain the product of the squares of 
 the differences of the roots aj, ag, ...a„ of (1). Employing the 
 usual notation for this product, viz., ^(ai, ag, ag, •••a„), we have 
 
 /(ai)/'(a2) -fM = i-iy '"'-'' PoH(a,, a,, a,, - a J ; 
 
 ... A = (-l)^^"-^^i>o''^-^r(ai, 02, a3, .•• a„). 
 
 II. The discriminant of an equation can also be obtained as 
 follows : 
 
 /(x) = 0, (1) and f(x) = 0; (2) 
 
 being simultaneous equations when f{x) = has equal roots, 
 the equation 
 
 nf{x)- xf'(x) = (3) 
 
 is also consistent with (1) and (2). Now (3) is an equation 
 of the (n— l)th degree; and finding the resultant of (3) and 
 f'(x) = 0, which is also of the {n — l)th degree, we obtain the 
 discriminant A as a determinant of order 2n — 2. For an 
 example, we shall find the discriminant of the cubic 
 
 PqX^ +PiX^-\-P2X +i>3 = 0. 
 
128 
 
 THEORY OF DETERMINANTS. 
 
 We have to find the resultant A of the equations 
 
 p^x^ + 2j92^' + 3^)3 = 0, 
 ^PqX^ + 2piX + p2 = 0, 
 
 p^ 2^92 3i)3 =0. 
 
 3jpo ^Pi P2 
 3po 2pi ^2 
 
 By the same process we find the discriminant of the biquad- 
 
 ratic 
 to be 
 
 P = 2h^ + ^Pi^ + GP2^ + 4^)3^ -\-P4 = 
 
 i^o ^Pi ^P2 Pa = 0. 
 
 Pq 32h ^P2 P3 
 
 po Spi 3p2 p-s 
 
 Pi 3^2 3^3 p^ 
 
 pi 3p2 3^3 i)4 
 
 2h 3^2 3|)3 i)4 
 
 This is accordingly the same as P — 27J^ = 0, where 
 
 / = P(,Pi - 4:p^ps 4- 3^2% 
 
 J = P0P2P4 + ^PiP22h - PoPi - Pi Pa - pi- 
 
 102. We may show that »/= is one of the necessary con- 
 ditions when the biquadratic P = of the preceding article has 
 three equal roots. Since 
 
 P=i)oa^ + 4_piar^-j-62)2aj' + 4i)3a^+P4 = (1) 
 
 * In many processes it is found more convenient to write a given func- 
 tion in the form of this equation, i.e., 
 
 + ^ (n — 1 ) p„_2 a:2 4- n;>«- 1 x + ;?„, 
 Z ! 
 
 in which each term is multiphed by tlie corresponding coefficient in the 
 
 expansion of {a:+l)*». Any given polynomial can, of course, be at once 
 
 reduced to this form. 
 
APPLICATIONS AND SPECIAL FORMS. 129 
 
 has three equal roots, two of these will be roots of 
 
 K^4-3i?iar^ + 3i>2a;+i)3 = 0, (2) 
 
 and one of them is a root of 
 
 Po^ + 2j9i 0^+^92 = 0. (3) 
 
 From (2) and (3) this root is also found in 
 
 Pix'^ + 2p.2X-^Ps = 0. (4) 
 
 Multiplying (3) b}* o^, (4) by 2x^ and adding, we obtain 
 ay'iPoX^ + 2piX H-i>a) + 2x{pixr + 2p.2X -fi^s) = 0. (5) 
 
 Now adding pa^ + '^Ih^ H~i^4 to the first member of (5), we 
 have, since P=0, 
 
 a^(PoX^-\-22h^-\-p2)-\-2x(piX^+2p2X+ps) -j^Paa^H- '^Ps^-\-p^= 0. 
 
 Hence, if (1) has three equal roots. 
 
 PqX^-\-2piX-{-P2 = 0, 
 PiX^-}-2p2X-\-ps = 0, 
 P2ic2+2j93aj+i>4=0. 
 
 or J= 0. 
 
 Po Pi Pi 
 
 Pi i>2 i>3 
 i>2 P& P4 
 
 = 0, 
 
 The other condition for three equal roots of (1) is accordingly 
 7=0. 
 
 103. The resultant of a system of n homogeneous equations, 
 one of which is of the second degree, and the remaining n—l 
 are linear, may be obtained as follows. Given 
 
 P=PQaf-\-p^y^--\-p2Z^-^2qoXy-\-2qiXZ-^2q2yz = 0, (1) 
 Pi= a^x -\-b^y +CiZ = 0, (2) 
 
 P2= a2X 4- bzy -\-C2Z = 0. (3) 
 
 Differentiating (1) with respect to x, ?/, z in succession, and 
 remembering Euler's theorem on homogeneous functions, we 
 obtain 
 
 P= x(poX-}-qoy-hqiZ) + y(qoX-\-piy-i-q2z) 
 
 + z(qiX-{- q2y -i-PiZ) = 0. (4) 
 
130 
 
 THEORY OF DETERMINANTS. 
 
 Equations (2) and (3) and (4) are simultaneous homoge- 
 neous equations ; hence, by 77, (4) must be expressible linearly 
 in terms of (2) and (3), and 
 
 ^1^1 + 192^2 = 
 
 (5) 
 
 is an equation identical with (4) . Equating the coefficients of 
 (4) and (5), we have the following system of equations: 
 
 Po« + go2/ + ^i2;-^iai- ^2^^2 = 0, "I 
 
 qiX-{-q2y+P2^—0iCi—02C2 = O. ) 
 
 Now, taking equations (2) and (3) with equations (E)^ we 
 have a system of five homogeneous equations. Eliminating 
 a;, y^ z, Oi, $2-, the resultant of (1), (2), (3) is 
 
 i2 = Pq go qi «i «2 
 go Pi g2 &i h 
 
 qi q-i P2 ci C2 
 
 tti bi Gi 
 a2 b2 C2 
 
 In general, let the system of equations be 
 
 f{x) =p,x,^ +P2X2 +P3X3 H ^PnXn' + ^qiXiX2 
 
 -{-2q2X^Xa-\ \-2q.^.x^_iX^ =0, 
 
 Pi =aiXi -hbiX2 -{-CiXs -{- "• -\-liX,, =0 
 
 P2 = aoXi +b2X2 -{-C2X^ +"'+kXn =0 
 
 P„ 1= a^.iXi-^b^_iX2-\-Cn-iXs-\ \-l 
 
 
 
 («) 
 
 We have, as before, if /x/ denote the differential coefficient 
 of f{x) with respect to fl;^, 
 
 a?iA' + ^2 A' 4- xja^j + - + Xnfx,! = 2/(0.') = 0. 
 
 (&) 
 
 Since (a) and (&) constitute a system of simultaneous 
 homogeneous equations, (6) considered linear with respect to 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 131 
 
 the variables, must be expressible linearly in terms of the 
 n — 1 linear equations of (a) . Hence (6) is identical with 
 
 Oll\ + e^P, + esPs+"'-h On-lPn-1 = 0. (c) 
 
 Equating the coefficients of (5) and (c), we obtain the 7i 
 homogeneous equations 
 
 qi^l +P2^2 + qn^S-\ \-q2n-l^n = Ml +^2^2 + MsH H K-lOn-1, 
 
 q^l + qn^2 + P-A^i-\ h q-ia-S^i = CA-^C202+ C3^3 H K C„_i^„_i, 
 
 g«-li»l + ^2«-ia^2 + ^3H-3^*3H f-Pni»»= ^A + ?2^, + ZAH h^H-A-l- 
 
 These equations, together with the n— 1 linear equations of 
 (a), form a system of 2n—l equations between x^, x.2, ••• x^, 
 Oil $2, •" ^n 1- Hence the resultant of the given system is 
 
 Pi 
 
 qi 
 
 ^2 
 
 qi 
 
 P2 
 
 qn 
 
 q2 
 qn 
 
 Ps 
 
 qn-l q2nl q-An-S 
 Qi 5i Ci 
 
 0,2 h.2 C2 
 
 a«^i ^„-l (^n I 
 
 qn-l «1 «2 
 
 q2u~l h h 
 
 q-Sn-ti Ci Co 
 
 Pn h k 
 
 Zl 
 
 /., 
 
 «H-1 
 
 Zn-1 
 
 
 
 
 
 
 
 
 Special Solutions of Simultaneous Quadratics. 
 
 104. By the help of a special expedient we may often solve 
 a pair of simultaneous quadratics much more rapidl3' and ele- 
 gantly with determinants than by the ordinary methods. The 
 following examples will serve to exemplify the method em- 
 ployed, and are, moreover, such forms as occur frequently. 
 
 A. Find x and y in 
 
 ttiic + ^i?/ _ mj \ 
 
 '~ ^ T, . (1) 
 
 ^ + f = 
 
132 
 
 THEORY OF DETERMINANTS. 
 
 Let / be such a factor that 
 
 
 (2) 
 
 From (2) 
 
 / 
 
 mi bi 
 nil 62 
 
 fD 
 A 
 
 / 
 
 y 
 
 «! mil 
 
 1 tti 62 1 ^ I «i ^2! 
 
 Substituting in the second equation of (1) 
 
 tD 
 
 fDi 
 
 A ■ 
 
 rA 
 
 B. Solve the equations 
 
 2/ = 
 
 rA 
 
 ±Vi>^+A' 
 
 ttiic + Wy — niixy 
 a^x -\- 622/ = maic?/ 
 
 (1) 
 
 Divide these equations member by member ; then, as before, 
 put 
 
 aiX-\-h^y = fmi^^ 
 
 a^x-^h^y^fm^]' 
 
 (2) 
 
 / I ^1 &2 I 
 
 y = 
 
 f I «i »% I 
 
 I ai 62 1 
 From the first equation of (1) 
 
 [a, I mi 62 I 4- &i I «i Wi2l] I «! &2 I 
 
 / = 
 
 £C = 
 
 mi I ?7li &2 I I «1 *^2 I 
 I tti 62 I I «1 ^2 
 
 I tti m2 1 
 
 y 
 
 I mi 69 I 
 
 A shorter solution is obtained by dividing each equation of 
 
 (1) bv icy, and solving for - and -. 
 
 X y 
 
APPLICATIONS AND SPECIAL FORMS. 
 C. Solve the equations 
 
 asX^ -h b2y^ = rriz ) 
 
 Write these equations 
 
 aiX +&i2/ = wi 
 a2X ' X -{- b2y ' y = m. 
 
 :! 
 
 133 
 
 (1) 
 
 (2) 
 
 
 
 mi 
 
 61 
 
 
 
 «! 
 
 m. 
 
 
 
 
 
 Then x = - 
 
 ma 622/ 
 A 
 
 ; 2/ = 
 
 a2X mg 1 
 A 
 
 (" 
 
 «! 6i 
 
 a2a; 622/ 
 
 We have 
 
 
 icA — Wi 62 2/ = — ma 61, 
 mia2a;+ A?/ = aim^, 
 ttg 61a? — ai 52.7 =— A. 
 
 
 Hence 
 
 
 
 
 
 
 A — mi^a ^2^1 
 
 = 
 
 0. 
 
 
 
 mittg ^ — aimg 
 
 
 
 
 
 ag^i — ai52 ^ 
 
 
 
 From which 
 
 
 A = ± Vc 
 
 
 
 
 11^63^2 + &l^ 0^2 ^2 
 
 -mi^a2&2* 
 
 Again, 
 
 
 dio; + biy =mi, 
 a26ia; — ai622/ = — ^• 
 
 
 
 mi 
 
 6. 
 
 
 ai mi 
 
 
 
 
 A 
 
 01*2 
 
 
 ttg^i —A 
 
 . (. 
 
 1 = 
 
 ai 61 
 cxa&i — ai62 
 
 "' = " A 
 
 ; y = 
 
 
 Ai 
 
 D. Solve the equations 
 
 QiX -\- biy = Ml I 
 
 «2aJ H- ^22/ + ^2^72^ =m2)' 
 
 These equations we write 
 
 aiX-\-biy =mi| 
 
 (1) 
 
 (2) 
 
134 
 
 THEORY OF DETERMINANTS. 
 
 \ nil h\ 
 
 
 A 
 
 As before, 
 
 Whence 
 
 02 + Cay h, 
 
 :!•) 
 
 (A + miCg)^/ — aim2+ mia2 = 0, 
 — hiC^y + tti 62 — (X2^i — A = 0. 
 
 A + mjCg mia2 — %^2 
 — 61 C2 (Xi^a — 0^2^! — ^ 
 
 = 0, 
 
 a quadratic from^which A is found. 
 
 I mi 62 1 
 
 y = 
 
 I Oi mg I 
 A -h m 1 C2 
 
 Example 5 above can also be solved by the method of this 
 example. 
 
 E. Solve the equations 
 
 aa? + hxy -\- cy^ = d } 
 ex^ -^ fxy -]- gy"- = h) 
 
 Equations (1) may be written 
 
 x^ -{- 2 aixy -{- biy^ = m 
 xP -\- 2 a2xy -\- biy^ = m^ 
 
 ;l 
 
 (1) 
 
 (2) 
 
 by easy reductions. We introduce the factor 2 for convenience 
 in calculation. A solution analogous to D could be given. 
 Whatever the coefficient of xy^ it can, of course, be at once 
 reduced to the form 2ai. We write equations (2) 
 
 x{x + a^y) -\r y {aiX-\- h,y) = m^ 
 x (x + a2y) + y (a.x -\- b. 
 
 hy) = mi I 
 ),y) = m2 j 
 
 (3) 
 
 Then 
 
 X = 
 
 Ml aiX-\-b^y 
 mo a^x-^-b^y 
 
 x-\-aiy mi I 
 X -\- a^y rrio \ 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 135 
 
 where 
 
 We have 
 
 Whence 
 
 x-\-aiy aiX-{-biy 
 x-\-a2y a2X-\-h2y 
 
 [A+ \aim2\~\x-{- I \m2\y = 0, 
 [ma — mi] a; + [ I «! mg I — A] 2/ = 0. 
 
 A + I aimal I ftimgl 
 
 ma — mi I a-^m^ I — A 
 
 0. 
 
 Solving this quadratic, 
 
 Now 
 
 A = ± Vl aimgP— 1 61 ms I {m^—m^, 
 
 \mMy 
 
 X = 
 
 A + I ai mg 
 
 Substitute tliis value of x in the first of equations (2), and 
 we have 
 
 I mi 62!^/ 2a^\mMlf 
 
 (AH- I aimgl)'^ A+Uim; 
 
 + \y^ = ^1, 
 
 a pure quadratic, from which the value of y can be found 
 at once. 
 
 105. To the solutions of the last article we add the follow- 
 ing, in which one equation is a quadratic and the other is a 
 cubic. 
 
 Find the values of x and y in 
 
 a? -{-xy -\-y^ mi 
 
 x^ — xy -\- y'^ ~~ m2 
 
 oc^-\-y^ = o? 
 
 (1) 
 
 From the first of equations (1) 
 
 • y = \mi I 
 .?/ = Am2 3 
 
 X {x -\- y) -\- y ' y = \mi 
 x(x — y)-\-y 
 
 (2) 
 
136 
 
 THEORY OF DETERMINANTS. 
 
 A 
 
 m, y 
 
 A 
 
 a; + 2/ mi 
 
 
 
 X = — 
 
 mo y 
 A 
 
 ; 2/ = - 
 
 x — ym^ 
 A 
 
 . (.. 
 
 a;-y y 
 
 We have 
 
 icA -- X (mi — wig) 2^ = | 
 
 
 Aa; (mi - mg) -f [A - A. (mi -f- mg)] 
 
 2/ = 0J 
 
 y.) 
 
 (3) 
 
 Whence 
 
 A A (?Hi — mg) 
 
 A(mi — ms) A — A(mi + m2) 
 
 = 0. 
 
 From this equation 
 
 ^ ~~ 2 ^^1 + ma ± Vl077iim2 — 3mi^ — 'dm/\ . 
 
 Now, since 
 
 A = 2/, 
 
 we have to find the vahie of A in order to complete the 
 sohition. 
 
 From equations (2), and the second of equations (1), 
 
 x-\-y 
 
 Am, 
 
 xy = - (wii - ma) 
 
 (4) 
 
 From equations (4) , and the first of equations (2) , we get 
 
 A = 
 
 Vi (3 mi m2 -r m^) 
 
 and hence 
 
 y = ± 
 
 (wi + Wo) ± VlOmjma— 3mi^— 3m.j 
 
 V ^ ( 3 m-i m2 — mi) 
 
 X may be found from the second of equations (1), or from 
 the first of equations (3) . 
 
APPLICATIONS AND SPECIAL FOKMS. 
 
 137 
 
 Solution of the Cubic, 
 106. The general cubic equation 
 
 PO^ -i-Pl^ +P2^ +i>3 = 
 
 is always reducible to the form 
 
 x^ + qiX + q2 = 0. 
 
 We are therefore only concerned with the solution of (2), 
 The determinant equation 
 
 = 
 
 is identical with 
 We have 
 
 (1) 
 
 (2) 
 
 X 
 
 «i 
 
 as 
 
 a2 
 
 X 
 
 ai 
 
 ai 
 
 a^ 
 
 X 
 
 a^ — Sa^ €12 x-\-ai^ -{- a/ = 0. 
 
 A= £c + ai4-a2 % (ig 
 x-\-ai-\-a2 X . ai 
 
 ^ + «1 + «2 ^2 ^ 
 
 (3) 
 
 hence a; -{- «! + ag is a factor of A. 
 
 Again, let a be one of the imaginary cube roots of unity ; 
 then the other is a^. Substitute aja, ag^^ foi" % and a^ re- 
 spectively in A, obtaining 
 
 2 /v» n ^ 
 
 since a*" 
 
 a2a' 
 
 A = 
 
 a^ — 3 aittga^^ + ^i^ + %^a^ = ^j 
 
 CC + aitt + a^a^ CKja 0^2^ 
 JC + Ofia + aoa^ » a^a. 
 a; rf «! a + agtt^ a2a^ ^ 
 
138 THEORY OF DETERMINANTS. 
 
 and hence A is divisible by ic + aia + aga^ By substituting 
 a^a? and a^a for a^ and a^ respectively in A, we obtain a 
 determinant A", which is shown equal to A in the same way 
 as before. 
 
 Hence a^a? -\- aza -\- x is also a factor of A, 
 
 Accordingly, 
 
 A = A: (a; + ai -f- <^2) (a? + otja + a^a^) (x + aia^ + aga) ? (4) 
 
 where A: is a numerical factor. Comparing the term a^ of A 
 with the term x^ in the second member of (4) , we see that k=l. 
 
 .*. x^ — 3ai a2X-\- cii + ag^ = (x-\-ai-\- a^ {x-^-a^a + a^ a?) 
 
 (ic + aja^ + aaa). (5) 
 
 From (5) we have at once 
 
 a; = — «! — as, — aia — aga^, — ^la^ — aga. 
 
 Now applying this result to the solution of (2), we put 
 
 gi = — Saittg^ ^2 = 0^1 +(^2 'i 
 whence 
 
 \ 2 ^ 4.^21 ' ^2\4^27 
 
 Hence, finally, the roots of (2) are 
 
 ^-f-#^ ^^-fWlVft'. 
 
 
 > 2 XT + S? 2 + > 2 + \T + 27 2 
 
APPLICATIONS AND SPECIAL FORMS. 139 
 
 Symmetrical Determinants. 
 
 107. When we regard the square of elements that make 
 up a determinant, it is natural to inquire what special proper- 
 ties, if any, the determinant possesses when we suppose the 
 elements not all independent ; in other words, what special 
 forms arise when we suppose certain relationships to exist 
 between the elements, and what are their most important prop- 
 erties. Among the special forms very frequently met with, 
 especially in Geometry, are the Symmetrical determinants. The 
 symmetry here referred to is first, symmetry with respect to 
 the diagonals^ and second, symmetry with respect to the inter- 
 section of the diagonals, i.e., the centre of the square. Two 
 elements, so situated that the row and column numbers of the 
 one are the column and row numbers of the other, are called 
 conjugate elements. Evidently the line joining two conjugate 
 elements a„ and a,^ is bisected at right angles by the principal 
 diagonal. If in a determinant a^^ = dsri then the determinant 
 is axisymmetric, or simply symmetrical. The definition of a 
 symmetrical determinant is extended so as to mean symmetry 
 with respect to the secondary diagonal also, so that a deter- 
 minant is symmetrical if for each element there is an equal 
 element so situated with respect to its equal that the line 
 joining the two is bisected at right angles by one of the diago- 
 nals. The following are symmetrical determinants : 
 
 ai bi Ci di 
 
 ? 
 
 a,i ai2 
 
 ai3 
 
 «14 
 
 ? 
 
 a. 
 
 &1 
 
 ^^ 
 
 d. 
 
 bi h,Xyd2 
 
 
 <Xi2 ^13 
 
 «14 
 
 ^24 
 
 
 ^2 
 
 62 
 
 C2 
 
 Cl 
 
 Ci ,.Cjj " < ds 
 
 
 a-13 ai4 
 
 ^24 
 
 «34 
 
 
 % 
 
 h 
 
 h 
 
 bi 
 
 di ^2 ds di 
 
 
 «14 ^24 
 
 «34 
 
 «44 
 
 
 ^4 
 
 as 
 
 ^2 
 
 ai 
 
 108. We have already had a number of problems which 
 gave rise to symmetrical determinants. The student may refer 
 to the last determinant in example IV., 84, to the first deter- 
 minant of 84, VII., to the form of the resultant obtained by 
 Bezout's method of elimination, 93, (I.), and to the value of 
 
140 
 
 THEORY OF DETERMINANTS. 
 
 «7, 102, for illustrations of how symmetrical determinants 
 occur in practice. Again, we have 
 
 Ittji CI22 ^33! — 
 
 ^11^21 "h Cl'12^22 "f~ ^13^28 ^11<^31 "f" ^12^32 ~f" <^13^33 
 
 0^21(^11 + Of 22^ 12 + ^23^13 Ct2i -\-Ct22 ~f"^23 
 
 ^21^*^31 ~l~ ^22%2 H~ <^'23^;i3 
 
 %l%l + <^32'^12 + ^33<^i3 <*31^2lH~%2^22 + <^33^23 ^31 + <^32^ + ^^33^ 
 
 which is obviously symmetrical. It is easy to show that the 
 square of any determinant is a symmetrical determinant. Let 
 
 then we have to show that b,s = ft^-r* 
 
 br, = a,ia,i + a,2a,2 + a,.3«,3 H + «rna.ni 
 
 bgr = ot^i a,.i + ag2 ci,.2 H~ ^«3 ^rs 4" • * • + <^,„ ofy^ ; 
 
 whence the proposition. An obvious coroUar}^ is that any even 
 power of a determinant is a symmetrical determinant. 
 
 109. It is evident that conjugate lines (a row and a column 
 having the same number) in a sj'mmetrical determinant are 
 composed of the same elements in the same order. Consider 
 now two minors M and Mi of any determinant such that the 
 rows and columns erased to obtain M are the columns and 
 rows erased to obtain M^. Then 
 
 ' M = 
 
 a^g 
 
 «/A 
 
 (ifi '" 
 
 , and Jfi = 
 
 %f 
 
 %9 
 
 %9 
 
 %n 
 
 «..• - 
 
 
 anf 
 
 O^ng 
 
 ... 
 
 ... 
 
 
 
 
 a,f 
 
 Clia 
 
 Now, if the determinant is symmetrical, so that a„ = a^,., 
 we have M= iV/i, and, in particular, A^s = -4,r ; or, in a sym- 
 metrical determinant, conjugate minors are equal. From this it 
 follows at once that the reciprocal determinant is symmetrical. 
 Further, it is evident that minors whose diagonal lies in the 
 principal diagonal of a symmetrical determinant (coaxial minors) 
 are themselves symmetrical. 
 
APPLICATIONS AND SPECIAL FORMS. 141 
 
 110. We may show that the product of a symynetric deter- 
 minant by the square of any determinant is a symmetric determi- 
 nant^ as follows : 
 
 Let 1 ain I be a symmetrical determinant, and put 
 
 I aiJ X I ai^l = I Ci„| , and I a^J x ki^l^ = \biJ. 
 Then 
 
 K = aaCa + f^i2Ck2 + afs^^s + *•• + ainCkn- (1) 
 
 In (1) substitute the values of c^, Cf,2i ••• C;^, and we have 
 
 + (Woia^i + a.2zaja + «23 «ft3 + • • • + <^2n ^^kn) (^i2 
 + 
 
 + («niaa + an2a;fc2 + «n3«A3 + "• + ««««&»)«£„, 
 
 = (^naa + Ct2ia,2 + «3ia;3 -!-•••+ «ni«in)aa 
 + («i2a<i + «22af2 + a32ai3 + ••• + «n2am)a*2 
 
 + 
 
 Since a^i, = a^^, this sum becomes 
 
 O-kl^il + aA2C,-2 + ••• 4- a^nCfn = hi' 
 
 Whence I biJ is symmetrical. 
 
 From this and 108 we see that any power of a symmetrical 
 determinant is a symmetrical determinant. 
 
 111. Cauchy's theorem for the expansion of a determinant, 
 example III., 63, assumes a somewhat different form when 
 the determinant is symmetrical. Thus, instead of 
 
 A = (loo A'- ^aioaokAi„, 
 we have, when A is symmetrical, 
 
 A = aoo A'- ^a^Q^Aa - 2 2a,oa;fco4-ft. 
 in which, as before, i has all integral values from 1 to w, and 
 
142 
 
 THEORY OF DETEEMIKANTS. 
 
 for ik we write the different combinations of the numbers 
 
 1,2, 
 
 taken two at a time. 
 
 For example, 
 
 a 
 
 h 
 
 9 
 
 h 
 
 b 
 
 f 
 
 9 
 
 f 
 
 c 
 
 a b c 
 
 a h g 
 
 b h f 
 
 c 9 f 
 
 ahc - ap - bg" - cJi" + 2fg7u 
 
 = (Tp J^h-g^-^e-W-'l abfg 
 = {af+bg-chy-4.abfg. 
 
 2acfk — 2bcgh 
 
 112. Consider the determinant 
 
 A = ai bi 
 
 d. 
 
 bi &2 ^2 <^2 62 
 
 Ci C2 C3 U3 63 
 
 dl d, dg ^4 64 
 
 61 62 63 64 65 
 
 and suppose that 
 
 «i+&i4-Ci+<^i + ei = 61+62+^2+5^2+62= Ci+Ca-f-Ca+cZg -1-63 
 = (^1+^2+^3+^^4+64 = ei+e2+C3+e4+e5 = 0. 
 
 Then, first, A = ; since, if we add tlie elements of the 
 other rows to the corresponding elements of the first row, the 
 elements of this row all vanish ; and, secondly, we can show 
 that all the first minors of A are equal. 
 
 B, 
 
 bi 
 
 Ci 
 
 d. 
 
 ei 
 
 C2 
 
 C3 
 
 ^3 
 
 63 
 
 d. 
 
 d. 
 
 d. 
 
 64 
 
 62 
 
 63 
 
 64 
 
 65 
 
 and (7o = — 
 
 «! 
 
 Ci 
 
 ^1 
 
 61 
 
 61 
 
 C2 
 
 d2 
 
 62 
 
 ^1 
 
 d. 
 
 ^4 
 
 64 
 
 ei 
 
 63 
 
 64 
 
 65 
 
 The first, third, and fourth columns of Bi are identical with 
 the second, third, and fourth rows of G^. By hypothesis the 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 143 
 
 elements of the first row of Cs are respectively — q, — Cg, — c?3, 
 — 63; whence 
 
 a = 
 
 A, 
 
 Ci C3 U3 63 
 
 Oj C2 Ct'2 62 
 
 di dg ^4 64 
 
 61 63 64 65 
 
 as was to be shown. 
 
 In general, if in a symmetrical determinant the sum of the 
 elements in each row is zero^ the determinant vanishes^ and all 
 the first minors are equal. 
 
 Let 
 
 A = I aoo«iia22 ••• a^J , with a„ = a,^, 
 
 and ttio + a.i + ai2 H + afn = 0. 
 
 That A vanishes is obvious. Again, 
 
 -^ 
 
 «11 
 
 ai2 
 
 ... 
 
 «]*-! 
 
 aU 
 
 «1A+1 
 
 «21 
 
 a22 
 
 ... 
 
 «2A-1 
 
 .a2A 
 
 <^2A+1 
 
 ... 
 
 ... 
 
 ... 
 
 ... 
 
 ... 
 
 ... 
 
 C^i-U 
 
 «i-12 
 
 ... 
 
 «i-lft-l 
 
 «i* 
 
 ^^i-lft-fl 
 
 aa 
 
 -ai2 
 
 \ 
 
 ttiA-l 
 
 «a 
 
 «tm 
 
 «i+ii 
 
 «H-12 
 
 
 ^j+l*-l 
 
 «t+l* 
 
 <^i+lft+l 
 
 ''nAi+l 
 
 «!„ 
 
 t+ln 
 
 To the ith row of Aq^ add the remaining rows ; the ^th row 
 becomes 
 
 Then to the Zcth column of A^ add the remaining columns ; 
 the hih. column becomes 
 
 Now, making the I'th row the first row, and the Ajth column 
 the first column, we have 
 
144 
 
 THEORY OF DETEKMINANTS. 
 
 A.= 
 
 
 
 
 
 
 
 
 (_!)<« 
 
 «oo 
 
 Ooi 
 
 ^02 
 
 
 «0*-l 
 
 <^0*+l 
 
 " «0» 
 
 
 aio 
 
 «u 
 
 ai2 
 
 
 <*!*-! 
 
 «!*+! 
 
 •• «!« 
 
 
 a* -10 
 
 «i-ii 
 
 <*i-12 
 
 
 «i-l*-l 
 
 <*<-l*+l 
 
 •• ai_ 
 
 
 «i+10 
 
 «t+ii 
 
 «i+12 
 
 
 <^i+U-l 
 
 Ctt+lA+l 
 
 " «i+ 
 
 
 «nO 
 
 «nl 
 
 Ct„2 
 
 
 «'nA-l 
 
 Cf'nk+l 
 
 - ««» 
 
 which pre 
 
 ves the theorem. 
 
 
 
 
 
 — -^1 
 
 113. If cc be subtracted from each element of the principal 
 diagonal of a symmetrical determinant, we have a function of 
 X which, equated to zero, gives an important equation. The 
 roots of this equation are all real, which may be proved as 
 follows. We have 
 
 /(«')= 
 
 an-x 
 
 aj2 
 
 ai3 
 
 
 
 am 
 
 
 «21 
 
 a22 — x 
 
 ^23 
 
 
 
 «2n 
 
 
 «31 
 
 aga 
 
 ^33- 
 
 X 
 
 
 «3n 
 
 
 ... 
 
 ... 
 
 ... 
 
 
 
 ... 
 
 
 «nl 
 
 «n2 
 
 ^nS 
 
 
 
 «n«- 
 
 Then 
 
 
 
 
 
 
 
 -x) = 
 
 an + a; 
 
 a,2 
 
 <^13 
 
 
 ... 
 
 «1« 
 
 
 ttsi 
 
 a^i + x 
 
 «23 
 
 
 ... 
 
 a2« 
 
 
 ttgi 
 
 %2 
 
 ^33- 
 
 X 
 
 ... 
 
 ttsn 
 
 «n3 
 
 = 0. 
 
 a„„ + a; 
 
 a« = a. 
 
 0) 
 
 (2) 
 
 Multiplying (1) 
 
 and (2), 
 
 
 
 /W/(-^) 
 
 
 
 
 = 
 
 Pn-0^ 
 
 P12 
 
 Pl3 "• 
 
 Pin 
 
 
 P21 
 
 P22-^ 
 
 P23 '" 
 
 P2n 
 
 
 Psi 
 
 P32 
 
 Pm-^ ... 
 
 Psn 
 
 
 Pnl 
 
 Pn2 
 
 PnS '•' 
 
 Pnn- 
 
 = 0, 
 
 Pr. =Pn- 
 
 (3) 
 
where 
 
 APPLICATIONS AND SPECIAL FORMS. 
 
 145 
 
 Expanding the determinant of (3) by 63, I., 
 
 \p,J-x'^D^_^-^x'^D^_,-x'^D^_,+ ... + {-^Y=0. (4) 
 
 Now i)„_i, D^^o^ Dn-s^ '"•> being coaxial minors of Ipinl, 
 are all sums of squares of minors of I aj^l ; for consider one 
 of these minors 
 
 i>n 
 
 Pff Pf9 
 Vgf Pgg 
 
 Prf Prg 
 
 ' Pfr 
 
 Prr P^t=Ptr 
 
 Z)„_2 iiiay 1>G obtained by squaring the array 
 
 */3 
 
 Vn 
 
 in which there are n columns and n — 2 rows. By 58, 1st, 
 Z>„_2 must be the sum of products of pairs of determinants 
 which in this case are equal ; hence D„_2 is the sum of squares 
 of minors of I ai„l of order n — 2. Hence 5Z>„_i, 22>„_2, 
 2Z>„_3, •••, are all positive. The signs of the terms of (4) are 
 therefore alternately positive and negative, and, by Descartes' 
 Rule of Signs (4), can have no negative roots. Accordingly, 
 /(a;) = 0, or (1), cannot have a root of the form aV— 1, for 
 then a? would be negative, which we have shown is impossible. 
 Nor can (4) have a root of the form /3+aV— 1 ; for if we 
 write au-"iS=au» a 22 — 1^ = ci' ^i etc., the proof just given 
 is applicable. 
 
 The student will find it interesting to apply the preceding 
 proof to the particular case where f{x) is of the third degree, 
 
 i.e. 
 
 /w= 
 
 a-x 
 
 • «12 
 
 «13 
 
 = 0, 
 
 «12 
 
 a22 + x 
 
 «23 
 
 
 «13 
 
 «23 
 
 ^33 + a; 
 
 Ctr. = «. 
 
146 
 
 THEOilY OF DETERMINANTS. 
 
 actually multiplying f{x) by /(—«), and expanding the result 
 to obtain the equation in a^, whose terms are alternately posi- 
 tive and negative. 
 
 114. Symmetrical determinants of the form 
 
 /6 
 
 «i 
 
 97 fe «i h 
 h 97 /e 
 
 d. 
 
 es 
 
 , and 
 
 Cs 
 
 d. 
 
 
 h 
 
 Cs 
 
 
 a, 
 
 h 
 
 
 fe 
 
 «! 
 
 
 ao 
 
 ai 
 
 a^ 
 
 ay 
 
 a., 
 
 ag 
 
 a^ 
 
 «3 
 
 ^4 
 
 ttg 
 
 a^ 
 
 a. 
 
 an -I a. 
 
 ^«+i 
 
 
 = P(aia2---a2„_2), 
 
 are called ortlio symmetric or persymmetric. That is to say, 
 when each line perpendicular to either of the diagonals has all 
 its elements alike, the determinant is persymmetric. Such' a 
 determinant can contain at most 2n — l distinct elements. 
 Examples of the occurrence of orthosym metric determinants 
 in practice are found in 84, VII. 
 
 115. The most important property of orthosymmetric deter- 
 minants is that the determinant remains unchanged when the 
 first terms of the successive orders of differences of its 2 n — 1 
 elements are substituted for the elements themselves. Consider 
 the following series of numbers, and form the 1st, 2d, 3d, 
 • •• (2n — l)th orders of differences by subtracting Oj^^i from a^ 
 
 throughout. Then adopting 
 
 the u 
 
 sual notation, v, 
 
 tto ai tta 
 
 ttg 
 
 «4 
 
 Ct5 '" Ci2n-2 
 
 Ai An 
 
 Ai2 
 
 Al3 
 
 Ai4 ••• Ai2„-3 
 
 A2 
 
 A21 
 
 A22 
 
 A23 ••• A2 2„-4 
 
 
 Ag 
 
 A31 
 
 Ago ••• A3 2„_5 
 
 
 
 A4 
 
 A41 ••• A4 2„-6 
 
 Aat-a 
 
APPLICATIONS AND SPECIAL FOEMS. 
 
 147 
 
 We now show that 
 
 A = 
 
 tto ao 
 
 tto Qo 
 
 €12 <^S 
 
 a„ a, 
 
 n+l "'/1+2 
 
 Ct'n-l 
 
 = 
 
 tto 
 
 Ai 
 
 A2 
 
 A3 
 
 •• A, 
 
 «n 
 
 
 Ai 
 
 As 
 
 A3 
 
 A4 • 
 
 •• A, 
 
 «u+l 
 
 
 As 
 
 A3 
 
 A4 
 
 A5 • 
 
 •• A, 
 
 ... 
 
 
 ... 
 
 ... 
 
 ... 
 
 ... . 
 
 .. . 
 
 «2n-2 
 
 
 A«_ 
 
 lA. 
 
 A„+i 
 
 An+2 
 
 .. A. 
 
 n+l 
 
 If in A the (n-l)th, (n-'2)th, ■ 
 from the nth, (n — l)th, (?i — 2)th, 
 we get 
 
 A = 
 
 ao 
 
 Ai 
 
 «! 
 
 An 
 
 «2 
 
 A12 
 
 column be subtracted 
 • column respectively, 
 
 ^ln-2 
 Aln-1 
 
 Am 
 
 ttn-l Ai„_i Ai^ Ai„+i ••• Ai2«_3 
 
 Repeating the operation successively, we obtain 
 A = 
 
 ao 
 
 Ai A2 
 
 A3 • 
 
 • K-l 
 
 «i 
 
 All A21 
 
 A31 • 
 
 • A,_ii 
 
 a-j 
 
 A12 A22 
 
 A32 • 
 
 • A._i2 
 
 ... 
 
 ... ... 
 
 ... ' . 
 
 
 
 a„_ 
 
 1 Ai,,_i A2„_i 
 
 A3n-1 • 
 
 • A„.i„_i 
 
 Operating in a similar manner upon the rows, we get 
 
 A = 
 
 Ai 
 
 Ai 
 
 A2 
 
 A3 
 A4 
 
 A4 
 
 A 1 A 
 as was to be shown. 
 
 A„+i A„+2 
 
 A„_i 
 
 A„ 
 A„+i 
 
 Thus 
 
 8 15 
 15 26 
 
 26 1 = 
 43 ! 
 
 15 26 43 68 
 26 43 68 103 
 
 3 
 
 5 
 
 2 
 
 2 
 
 5 
 
 2 
 
 2 
 
 
 
 2 
 
 2 
 
 
 
 
 
 2 
 
 
 
 
 
 
 
 2' 
 
148 
 
 THEORY OF DETERMINANTS. 
 
 for we have 
 
 Similarly, 
 
 3 8 15 26 43 68 103 
 
 5 7 11 17 25 35 
 
 2 4 6 8 10 
 
 2 2 2 2. 
 
 7 0-4 
 
 — 
 
 ■5 
 
 = 
 
 7 
 
 -7 
 
 3 
 
 
 
 = ( 
 
 -4 -5 -3 
 
 
 -7 
 
 3 
 
 
 
 
 
 
 _4 _5 -3 2 
 
 
 3 
 
 
 
 
 
 
 
 
 -5 -3 2 10 
 
 
 
 
 
 
 
 
 
 
 
 dent may show that 
 
 
 
 
 
 
 12 4 7 
 
 =: 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 2 4 7 12 
 
 
 1 
 
 1 
 
 
 
 1 
 
 
 
 4 7 12 19 
 
 
 1 
 
 
 
 1 
 
 -2 
 
 
 
 7 12 19 30 
 
 
 
 
 1 
 
 -2 
 
 5 
 
 
 
 1 4 9 16 
 
 = 0. 
 
 
 1 
 
 8 
 
 27 
 
 64 
 
 4 9 16 25 
 
 
 
 8 
 
 27 
 
 64 
 
 125 
 
 9 16 25 36 
 
 
 
 27 
 
 64 ] 
 
 125 
 
 216 
 
 16 25 36 49 
 
 
 
 
 64 
 
 125 i 
 
 >16 
 
 343 
 
 = 6^ 
 
 Besides exhibiting obvious simplifications, these examples 
 show that wlien the elements of a persymmetric determinant 
 of the nth degree form an arithmetical progression of order 
 m* < n — 1 , the determinant vanishes ; and if the order of the 
 progression is n— 1, the determinant reduces to an nth power. 
 
 *The 
 
 series of numbers 
 
 
 
 
 
 
 
 1 
 
 8 
 
 27 
 
 64 
 
 125 
 
 216 
 
 form an 
 
 arithmetical progression of the third order, because the terms of 
 
 the third order of differences are alike. 
 
 
 
 Thus 
 
 
 
 
 
 
 
 
 1 
 
 8 
 
 27 
 
 64 
 
 125 
 
 216 
 
 
 
 7 
 
 19 
 
 37 
 
 61 
 
 91 
 
 
 
 
 12 
 
 18 
 6 
 
 24 
 6 
 
 30 
 6. 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 149 
 
 116. The conditions of the last statement will always be 
 fulfilled if a^ is a rational integral function of k of the mth 
 degree, whose highest term has the coefficient 1. For then, 
 according to the well-known theorem, a^^ aj, ag, ••• form an 
 arithmetical series of the mth order, of which the mth. dif- 
 ferences will be m!. If, then, ?7i = n — 1, all the elements of 
 the secondary diagonal will be (n — 1)!, and all the elements 
 below it will be zeros. Whence the determinant equals 
 
 (_l)2(«-i)[(^ _!)]«. 
 
 If m < n — 1, the determinant of course vanishes. In either 
 case, instead of ao, ai, ag, •••, we may write 
 
 If, for example, p is any given number, and 
 
 -\-m\ _{p-{-k + m){p-\-k-\-m — l) ••• {p-\-k-{- 1) 
 
 \ m 
 
 A) + mN /p -f m -f 1 
 
 \ m ) \ m 
 
 A)-hm + l\ /p + m-f2\ 
 
 \ m J \ m J 
 
 'p-\-2m 
 
 m 
 
 ^p + 2m\ A) + 
 
 2m + l 
 m 
 
 et.'") 
 
 117. Consider the determinant 
 
 A 
 
 k 
 
 kr 
 
 ki"" . 
 
 .. A;r"-^ 
 
 kr 
 
 kr' 
 
 ki^ . 
 
 .. kr- 
 
 kr' 
 
 ki^ 
 
 kr"" . 
 
 .. kr--^' 
 
 ^y.n-1 ^.^n J^^+l ... ^^n^ 
 
150 
 
 THEORY OF DETERMINANTS. 
 
 whose elements are in geometrical progression. That A must 
 vanish is obvious at sight ; for dividing any column except the 
 first by the ratio ?*, A is seen to contain identical columns. 
 Hence if the elements of a persymmetric determinant form a 
 geometrical progression, the determinant vanishes. 
 
 118. To the results of the last article we add the following. ' 
 Suppose in 
 
 A = 
 
 ^2 
 
 
 Cfn-l Clr 
 
 ^n+1 
 
 each element divides every other element whose subscript is 
 higher than its own, i.e., in general, 
 
 a^ = 6o 6i 62 • • • 6^. 
 
 Then 
 
 
 
 
 A = 
 
 60 
 
 6061 
 
 606162 
 
 6061 
 
 606162 
 
 bobib^bs 
 
 606162 
 
 60616263 
 
 6061626364 
 
 60616263 
 
 6061626364 
 
 60616.636465 
 
 6o6i62---6^_i 6o6i62---6„ 606162 •••6„.f.i bob^b.^-'-b^^z 
 
 6o6i62---6„_i 
 6061 "'b^_ib^ 
 bobi •••6„6„+i 
 
 Now it is obvious that 60 is a factor of the first row of A, 
 6061 is a factor of the second row, 606162 is a factor of the 
 third row, and so on. Hence 
 
 A=n br^ 
 
 i=0 
 
 1 
 1 
 
 1 
 
 b, 
 b, 
 bs 
 
 bibo 
 bibs 
 b,b. 
 
 616263 
 626364 . 
 636465 
 
 6162 •••6„_i 
 6263 •..6„ 
 • b,b, ...6.^1 
 
 1 
 
 bn 
 
 bnbn+l 
 
 Kbn+lbn+2 • 
 
 • bX^i-b.^^ 
 
APPLICATIONS AND SPECIAL FOKMS. 
 
 151 
 
 Skew Determinants, and Skew Symmetrical 
 Determinants. 
 
 119. We have heretofore shown (108) that the square of 
 any determinant is a symmetrical determinant. If we now 
 write the determinant of even order 
 
 3 
 
 ?>! 
 
 Cl 
 
 ck 
 
 = 
 
 «2 
 
 b. 
 
 Ca 
 
 d. 
 
 
 «3 
 
 h 
 
 C3 
 
 ds 
 
 
 a^ 
 
 h 
 
 C4 
 
 ch 
 
 
 h 
 
 ai 
 
 -d. 
 
 Cl 
 
 h 
 
 02 
 
 -d^ 
 
 C2 
 
 h 
 
 «3 
 
 -d. 
 
 C3 
 
 h 
 
 a^ 
 
 -d. 
 
 C4 
 
 we get, by multiplying these factors together, 
 
 ' - (^162) - (M2) -(«A)-(M3) -(«A)-(M4) 
 
 (« A) + (M2) ; - {(^ih) - {C2d3) - {ciM - (c^d,) 
 
 («A) + (M3) '(«2&3)-f (^2(^3) — («A) — (^3^4) 
 
 (aA) + {cA) ■ (a^b^) + (02^4) (a^b^) + (c^d^) 
 
 \ . • 
 
 In this determinant each element is equal to its conjugate 
 
 with opposite, sign, and the elements of the principal diagonal 
 
 are zeros. Such determinants are called skew symmetHcal. 
 
 In other words, if in a determinant we have a,^ = — a^i and 
 
 af^ = 0, the determinant is skew symmetrical. If an is not 
 
 zero, we have a skew determinant. It may be shown that the 
 
 square of any determinant of even order can be expressed as 
 
 a skew symmetrical determinant. Thus, since 
 
 A = 
 
 ttll 
 
 «12 
 
 
 (Xl3 Oi4 
 
 «21 
 
 ^22 
 
 
 ^23 «24 
 
 ... 
 
 ... 
 
 
 
 
 «n-ll 
 
 ««- 
 
 12 
 
 ttn-lS «n-14 ••• 
 
 «nl 
 
 «n2 
 
 
 «n3 a«4 ••• 
 
 ^n-ln-3 On^in-2 ^n-ln-1 ^^n-ln 
 <^nn-3 <^n»i 2 <^nn-l <^;m 
 
152 
 
 THEORY OF DETERMINANTS. 
 
 ^22 — ^21 ^24 — ^23 
 
 ^n-12 
 Ctn2 
 
 ^ln-2 — ^ln-3 ^In — ^n-1 
 <^2»-2 — ^2»i-3 ^2» — ^n-1 
 
 ^n-ln-2 — <^n-ln-3 ^'n-ln — <^«-ln-l 
 
 we have, after multiplying these determinants together, 
 
 ' A2 = 
 
 nn—l 
 
 
 
 mi2 
 
 Wi3 . 
 
 •• mi„ 
 
 msi 
 
 
 
 ^23 • 
 
 • W'2n 
 
 mgi 
 
 WI32 
 
 . 
 
 • Wig, 
 
 m« 
 
 m. 
 
 m„ 
 
 m,. = — m. 
 
 For 
 
 m,4 = tt,ia;(.2 — ^i2^kl + ^!3^ft4 — <^!4<^>t3 4- ••• + Otfn-l<^*n ~ <^in ^*n-l? 
 
 and hence m^i = 0, and mi;^ = — '^h* 
 
 120. The consideration of skew determinants reduces to 
 that of skew symmetrical determinants, as we shall now show^ 
 
 I. By 47, 
 
 A^") 
 
 <^21 ^22 
 
 Ct2n 
 
 Of' JIM. 
 
 a,^ = 
 
 Now, since a^j = — a;^;, the determinants Aq^''^ Aq^''"^^, Ao^'*""^^ 
 • ••, are all skew sj'mmetrical, and A^"^ is expressed in terms 
 of skew symmetrical determinants. 
 
 II. If, further, a^ in A^"^ is equal to a;, we have 
 
 It will soon be shown that a skew symmetrical determinant 
 of odd order vanishes. Accordiugl}', the terms of this expan- 
 sion in which the degree of Ao is odd will vanish. Thus 
 
APPLICATIONS AND SPECIAL FOEMS. 
 
 153 
 
 + x 
 
 + 3^ 
 
 
 X —a 
 
 - 
 
 -h — c 
 
 = 
 
 
 
 
 a X —d —e 
 
 
 a 
 
 
 b d X -f 
 
 
 b 
 
 
 c e f X 
 
 
 c 
 
 -d -e 
 
 + 
 
 -h -c 
 
 + 
 
 d -f 
 
 
 h -/ 
 
 
 e 
 
 f 
 
 
 f 
 
 1 
 
 
 -a 
 
 -b 
 
 — c 
 
 
 
 -d 
 
 — e 
 
 d 
 
 
 
 -/ 
 
 e 
 
 / 
 
 
 
 —a 
 
 -c 
 
 + 
 
 a 
 
 — e 
 
 
 c e 
 
 
 
 
 
 
 —a 
 
 -b 
 
 
 a 
 
 
 
 -d 
 
 
 b 
 
 d 
 
 
 
 
 r 0-/ 
 
 + 
 
 -e + 
 
 -d 
 
 + 
 
 -c 
 
 + 
 
 -b 
 
 + 
 
 -a ~ 
 
 L/ 
 
 
 e 
 
 d 
 
 
 c 
 
 
 b 
 
 
 a _ 
 
 = aj' + (/' + e2 + cZ2 + c2 + 62 + a^) a^ + (ct/*- 5e + cd)\ 
 The student may show that 
 
 = (a2 + 62 + c2 4-^2)2. 
 
 a 
 
 6 
 
 c 
 
 d 
 
 -6 
 
 a 
 
 -c? 
 
 c 
 
 c 
 
 d 
 
 <x 
 
 -b 
 
 —d — c 6 a 
 
 Writing another skew determinant A^, whose elements are 
 e, /, g, h, in the same form as A just written, we see that 
 Ai= (e2-|-/2 + fy2 _|_ 7^2)2^ jf ^g multiply A and Ai together 
 
 by rows, we get another skew determinant Ao, of the same 
 form as A and Aj ; the value of Ag may accordingly be written 
 
 where 
 
 m 
 n 
 
 (m2 + 7^2 +02+p2)2^ 
 
 ae 4- 6/-f eg + dh, o 
 — af-\- be — di + c?^, ^ 
 
 ag + 67i + ce — d/, 
 a/i — 6^ + c/4-de. 
 
 We have then 
 
 AAi = (a2+ 62 + C2+ ^2)2 (e2 + /2^ ^2^ ;^o^o _ ^^^o_^ ^2^ ^2 _^^2)2^ 
 or (a2 + 52 _|_ c2 + c?2) (e2+/2^ ^2_|_ 7^2^ ^ ^^^2 + ^2 _|. ^2 _|_^2)^ 
 
 which is Euler's theorem. 
 
154 
 
 THEORY OF DETERMINANTS. 
 
 121. Returning now to the consideration of skew symmet- 
 rical determinants, let us take the two minors M and Mi of 
 109, and making a^^ = — a^^^ an = 0, the determinant itself is 
 skew symmetrical ; M becomes M\ and M^ becomes Mi. 
 Now since every element of Jf' equals each element of Mi 
 with contrary sign, or since 
 
 Jf' 
 
 and Ml 
 
 -a, 
 —a. 
 
 where m, as before, is the order of the minors, i.e., the con- 
 jugate minors of a skew symmetrical determinant are equal if 
 m is even ; but if m is odd, the conjugate minors are equal, with 
 contrary signs. 
 
 In particular, if n is odd, Aij, — A^^. 
 
 But if n is even, Aij, = — A^i. 
 
 122. If the skew symmetrical determinant 
 
 A = 
 
 
 
 ai2 
 
 
 -ai3 
 
 is multiplied by ( — 1)'', we obtain 
 
 -A= 
 ai2 
 
 But since the rows of A are the columns of 
 
 A = -A, or A = 0. 
 
 -ai2 
 
 -ai3 
 
 
 
 -<*23 
 
 «23 
 
 
 
 A, 
 
 It is obvious that, in general, the effect of multiplying a 
 skew symmetrical determinant A of order n by ( — 1)" is to 
 change the rows into columns. Hence, when n is odd. 
 
 A=- A. 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 155 
 
 Therefore a skew symmetrical determinant of odd order 
 vanishes. In a skew symmetrical determinant An is, of course, 
 skew symmetrical ; hence 
 
 123. From 121, where n is odd, the reciprocal determinant 
 is symmetrical ; and, if n is even, the reciprocal determinant is 
 skew symmetrical. 
 
 124. I. Consider the following determinant 
 
 A = 
 
 
 
 ^14 
 
 
 
 ass 
 
 «24 
 
 -ai3 
 
 -«23 
 
 
 
 ^34 
 
 -«14 
 — «24 
 
 
 
 and the reciprocal determinant 
 
 Now, by 61, 
 
 
 
 — -^13 
 
 -Au 
 
 
 -A. 
 
 
 
 — ^24 
 
 -^13 
 A2S 
 
 
 
 -^34 
 
 
 
 = A 
 
 A = 
 
 A24 
 
 A^ 
 
 
 
 I ^ 
 
 <^2S 
 
 
 .-. ^1/ =j= a23^A, or 
 
 and hence A is a perfect square. 
 
 II. We shall now show that, in general, a skew symmetrical 
 determinant of even order is a perfect square. 
 
 Let 
 
 A = 
 
 
 
 «21 
 
 ai2 
 
 
 • «1» 
 
 • ^2n 
 
 . 
 
 *2 n+1 
 
 t-nn+l 
 
 ''»H-11 "'M+12 
 
 ''n+ln 
 
 <^2n-l 1 ^2«-l 2 • ' * <^2n-l n ^hn-l n+1 
 ^2n 1 <*2» 2 • • * <^2n n ^2» n+1 
 
 <*12n-l 
 ^2 2»-l 
 
 
 
 ^2/» 2n- 
 
 %2» 
 <^2 2n 
 
 ^2n-l 2n 
 
 
 
 ait= 
 
156 
 
 THEORY OF DETERMINANTS. 
 
 Then, as above, 
 A 
 
 Aa... , A, 
 
 = A . Aa. 
 
 11> <»2n2n« 
 
 (a) 
 
 ^2n 1 <*2n 2« 
 
 Now since A is skew symmetrical, and n is even, 
 
 ^aii 
 
 ^^«2.1=^ 
 
 «2«2n 
 
 A, 
 
 ; and A 
 
 ^12« 
 
 -A 
 
 «2nl' 
 
 «11» «2«2/i ' 
 
 or A = 
 
 <hnl 
 
 (6) 
 
 Therefore A is a perfect square if ^au,a2n2n is a perfect 
 square. In other words, a skew symmetrical determinant of 
 order 2?i is a perfect square if one of the next lower even order is. 
 But it is obvious that a skew symmetrical determinant of the 
 second order is a perfect square, and we have shown above 
 in I. that one of the fourth order is a perfect square ; hence, 
 by what we have just proved, a skew symmetrical determinant 
 of the sixth order is a perfect square, and so on. Hence the 
 theorem is true universally. 
 
 For a simple illustration, let us apply (6) to the following 
 determinant : 
 
 A = 
 
 = (vx — uy-\- tzf. 
 
 
 
 — X 
 
 X 
 
 
 
 y 
 
 t 
 
 z 
 
 u 
 
 -y 
 
 —z 
 
 = 
 
 —X —y —z 
 
 —t 
 
 — u 
 
 
 —t —u 
 
 
 
 — V 
 
 
 
 
 t —V 
 
 V 
 
 
 
 -t 
 
 
 
 
 
 t 
 
 As another application, we establish the followitig relation : 
 
 9 {a^—a^y {a^—a^Y («3— ^i)^ («2— «4)^ («i— "2)^ (as— «4)^ 
 
 The first expression equals (see example 7, page 37) 
 
 a,' 
 
 a,^ 
 
 tti 
 
 1 
 
 X 
 
 1 
 
 — 3% 
 
 3ai2 
 
 -a,' 
 
 ai 
 
 a/ 
 
 a^ 
 
 1 
 
 
 1 
 
 — 3^2 
 
 3 a/ 
 
 -a/ 
 
 ai 
 
 a/ 
 
 ttg 
 
 1 
 
 
 1 
 
 -3a3 
 
 3 as' 
 
 -os^ 
 
 a,' 
 
 a,' 
 
 «4 
 
 1 
 
 
 1 
 
 — 3a4 
 
 3a,2 
 
 -«/ 
 
APPLICATIONS AND SPECIAL FORxMS. 
 
 157 
 
 
 
 (a^-aiy 
 {a,-a,y 
 (a,-a,y 
 
 {a^-a^y 
 
 
 {a.,-a,y 
 (a^ - a^y 
 
 (oi - a^y 
 
 
 
 {a^-a^y 
 
 {a^-a^y 
 
 {0-2- a^y 
 
 
 
 {a^-a^y {ai-a,y 
 
 {a^ — a^y^ {a^ — a^y 
 
 {a,-a,y 
 {a^ — a^y 
 
 {a^-a^y 
 (a2-a^y 
 ias-a,y 
 
 -^ («2-«3)^ 
 
 = [(a2-«3)^ (ai-a4)3+ (a^-aj)\a2-a^y-\- (a^ -a2y(as-aiyf, 
 
 as was to be shown. 
 
 125. The following proof of the preceding theorem has some 
 advantages over the one just given. Let A be a skew symmet- 
 rical determinant of even order. Then A^^^ vanishes. Let 
 pij, be the complementary minor of an, in A^^^, and hence a 
 second minor of A. By 60, 
 
 
 = 0; 
 
 and since 
 
 ft-. = ft., PuPuu=-PJ' 
 
 (1) 
 
 (2) 
 
 Expanding A by Cauchy's theorem, 63, III., in terms of the 
 elements of the first row and first column, we have, since 
 
 ^Oll 
 
 0, 
 
 A = — ISttif ttaft* = ^(^li^ik '^^iil^kk') substituting from (2) , (3) 
 
 in which i*, k have the values 2, 3, ••• 2n. From (3) we have 
 at once 
 
 A = [2ai, Vft]^ 
 
 Here A is expressed as the square of a linear function of 
 the elements of the first row. This function is rational if 
 Vfti is rational. But pa is a skew symmetrical determinant of 
 order 2?i — 2. Hence a skew symmetrical determinant of order 
 2n is a perfect square if one of order 2w — 2 is. But we 
 proved (124, I.) that a skew symmetrical determinant of the 
 
158 
 
 THEORY OF DETERMINANTS. 
 
 fourth order is a perfect square ; hence, by what we have just 
 proved, one of the sixth order is a perfect square, and so on. 
 
 126. Since 
 
 that is, since A is the square of a linear function of the ele- 
 ments of the first row, we see that if A is of the fourth order, 
 Va contains 3 terms ; then, if A is of the sixth order, Va 
 contains 5.3 terms, etc. In general, then, Va is the sum of 
 
 (2n-l) (271-3) ... 5. 3. 1 terms. 
 
 Every term of Va is, moreover, the product of n elements of 
 A, in which no subscript is repeated. For, taking the term 
 ^14 V^44, for instance, we see that it consists of terms in which 
 neither of the subscripts i, 4 is repeated. But VAw will contain 
 a term a23 VTss' ^^ which, as before, -^/y^ contains none of tbe 
 subscripts 1, 2, 3, 4 ; find so on. Hence Va is the sum- of 
 terms of the form 
 
 ai2 0^34 ^56 • * * <^2n-12n1 
 
 in which no subscript is repeated. 
 
 If A is of the fourth order, for example, we have 
 
 AW = 
 
 ai2 ai3 a]4 
 
 C*2i v/ tt23 0^24 
 
 (I3I €ts2 0^34 
 
 «41 ^42 «43 
 
 , and Va^) = {ai^a^ ± ai^a^i ± auO^) • 
 
 diu = — a. 
 
 To determine which sign is prefixed to each term, we observe 
 that since the interchange of two subscripts of A amounts to 
 an interchange of two rows, and also of two columns, and 
 therefore leaves A unchanged, Va must be a function in which 
 the interchange of two subscripts either causes no change or 
 simply a change in sign. 
 
APPLICATIOiSrS AND SPECIAL FORMS. 159 
 
 If we consider any term of Va^''^ as ai2«34, which the inter- 
 change of the subscripts 1,2 transforms into a2iasi = — ai2a^, 
 it is obvious that Va^^^ does change sign on interchanging two 
 subscripts. We have then the square root equal to 
 
 ^12^34 — ^13^24 ~l~ ^14^23 5 (2) 
 
 for, if the second term of (2) were +, the interchange of 2 
 and 3, while changing the sign of the last term, leaves the 
 signs of the first two unchanged. 
 
 Since «,*= — ot^i, it is alwajs possible to so interchange the 
 subscripts that all the terms shall be positive. Thus 
 
 Va<^> = ai2a34 + ai3a42 + a^aas. 
 
 127. In general, we proceed as follows : 
 A being a skew symmetrical determinant of the 2 nth order, 
 A contains the term 
 
 ( — I) ^'12<^21<^34<^43^56<^*65 ••• ^2n-l 2» ^2n 2n-l = (<^12<^34<^56 * * * ^2n-12n) • 
 
 Hence Va contains the term 
 
 The positive square root of A which contains T as its first 
 term is an important function, possessing many properties 
 analogous to the properties of determinants, and is called a 
 Pfaffian. The notation 
 
 P= [1, 2, ... 2n], or (1, 2, 3, .•• 2w), 
 
 has been adopted for the Pfaffian. From what precedes, we 
 see that the terms of the Pfaffian are obtained from the prin- 
 cipal term by permuting the subscripts 2, 3, ••• 2n in all pos- 
 sible ways, and changing sign with every permutation. 
 
 Since a,j = — a^^,-, we may so arrange the elements that every 
 term of P is positive. Thus in the case of A^^^ above we have 
 
 Va(^ = P = ai2 a34 -f ai3 a^^ + a^ «23- (i>) 
 
160 THEORY OF DETERMINANTS. 
 
 128. If two subscripts are interchanged, the sign of P is 
 changed. Let a^,/3 be the terms of P containing the element 
 a^g. Then the elements of /? do not involve the subscripts 
 r and s. Interchanging r and s, let P become P'. Now 
 
 P^=^P'\ 
 
 since each square is A, in which two rows and also two columns 
 have been interchanged ; 
 
 ,\P = ±PK 
 
 But because of the interchange in r and s, 
 
 ays(^ becomes —a^,/3', 
 
 or, since the term a,„/3 of P = — a^^f3 of P', it follows that 
 P=2 — P\ as was to be shown. 
 
 129. We shall now prove a theorem by which we may com- 
 pute Pfaffians of order 2n from those of order 2n — 2.* 
 
 Assuming 
 
 VS=(-1)'(2, 3, ..., t-1, z + 1, ..., 271), (1) 
 
 or, after making i — 2 cyclical interchanges, 
 
 VS=a+l, 1 + 2, ..., 271, 2, 3, ..., ^-l), (2) 
 
 where /? has the same meaning as in 125, we show that 
 
 VSV/y^ = ft,; (3) 
 
 and then since 
 
 P = ai2 V^22 + «13 V/?33 H f- ai2n V/?2«2«5 (4) 
 
 * There is a difference in the nomenclature. We have here considered 
 the order of the Pfaffian to be determined by the number of subscripts 
 involved. Some authors determine the order of the Pfaffian by the order 
 of the terms in the elements. Thus (1, 2, 3, 4), or j \an\, which we have 
 designated as a Pfaffian of the fourth order, is said by some writers to be 
 of the second order. 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 161 
 
 (1, 2, 3, ..., 2 w) = a,, (3, ..-, 2n) + a^ (4, ..., 2n, 2) + - | .5) 
 + ai2«(2,3, ...,2n-l) j 
 
 To show that upon the assumption (1) or (2) the equation 
 (3) results, we proceed as follows : 
 
 Since 
 
 PiiPkk — Hik > 
 
 the terms of V/Sii V/S^^i must be equal each to each to the terms 
 of ^ai or equal with contrary sign. The product 
 
 (-l)'+^(2, 3 
 (2 
 
 , 3, ..., i-l, ^ + l, .-., 2n) | .g. 
 
 , 3, ..., k-1, A; + l, •••, 2n) j ^ ^ 
 
 becomes, after a certain number of interchanges, 
 
 where p, q, r, •••, u, v denote the series of numbers 
 
 2, 3, ..., 2n, 
 exclusive of i, k. Again, 
 /8«=(-l)'-^' 
 
 (7) 
 
 ^2 2 
 
 "1-12 ^i-13 
 
 t't+i 2 "i+i ; 
 
 ^2A-l 
 
 S+l Ai-1 
 
 ^2k+l 
 ^3k+l 
 
 i-1 A;+l 
 i+1 *+l 
 
 (8) 
 
 becomes, after the same number of interchanges as were em- 
 ployed to change (6) to (7), 
 
 (9) 
 
 a^kp 
 
 «*, 
 
 a*r 
 
 " a,. 
 
 «« 
 
 %P 
 
 a,. 
 
 a^ 
 
 " %v 
 
 Si 
 
 a,. 
 
 «.. 
 
 a,. 
 
 " a^v 
 
 a,i 
 
 ... 
 
 ... 
 
 ... 
 
 
 
 ... 
 
 a^ 
 
 a.2 
 
 a,r 
 
 •• Clvv 
 
 avi 
 
162 THEORY OF DETERMINANTS. 
 
 Now the first term of the product of (7) is 
 
 which is identical with the first pterin of the determinant (9). 
 Whence the truth of (2) is established, and (5) gives the 
 desired expansion of P. It is to be noted that the successive 
 terms of P are written cyclically. For example, A^*^ being a 
 skew symmetrical determinant of the fourth order, 
 
 AW = P^= (1,2,3,4)2, 
 and (1,2,3,4) = a-^a^^ + a^^a^ + ai4«23- 
 
 A(6) = p2^ (1,2, 3, ..-6)2, 
 
 (1, 2, ... 6) = ai2 (3, 4, 5, 6) + a,, (4, 5, 6, 2) + a^ (5, 6, 2, 3) 
 
 + ai5(6,2,3,4) + ai6(2,3,4,o) 
 
 = 0ti2 Cfc34 (Xgg + fti2 <^35 (^64 "f" ^^12 ^36 ^45 
 + «13 «45 <*62 4- ttjg a46 (X25 + ^13 Of 42 ttsg 
 + ai4 0t56«23 + «14 0t52«36 + ai4«53«62 
 4- a 15 a62 ^34 + «15 «63 «42 + «15 «64 «23 
 + aiett^sa^ + «16«24«53 + «16«25«34. 
 
 130. The student must have already noticed the analogy 
 between determinants and Pfaffians referred to above. The 
 following notation, based upon this analogy, is interesting. 
 Since the Pfaffian involves just half the elements of a skew 
 symmetrical determinant like A of 124, II., we write the 
 Pfaffian 
 
 P =: \ (li2 ^13 ^14 *•• ^1 2n 1 ^I2n 
 
 ^23 0^24 ••• 0^2 271-1 ^2 2n 
 
 0^34 ... ^3 2/1-1 Of3 2n 
 
 (hn-2 2n 1 ^2n-22n 
 ^2»-12n 
 
 which is shortened to 
 
 1 1 «12«23«34 • • • «2r» -12« | , Or tO ff{a^ 2») 5 Or tO 1 1 ttj 2n 
 
APPLICATIONS AND SPECIAL FORMS. 163 
 
 In particular, we have for a Pfaffian of the third order 
 
 Ui bi Ci =ff{aib2Cs) = \\ai h^ C3I 
 
 We may accordingly write equation {p), at tlie end of 127, 
 
 Vaw= ||ai4|, or rather A^^^ = \\a^^f ; 
 and the general equation would be 
 
 131. We must here conclude the discussion of Pfaffians with 
 the theorem: a bordered* skew symmetrical determinant is 
 the product of two Pfaffians. 
 
 From equation (6), 122, II., 
 
 ^ «2nl — ^ ' '^«ll.«2n2n' 
 
 /. Aa,„i=(l,2,...,27i)(2,3, ...,2n-2, 2n-l), (1) 
 
 which proves the theorem when the determinant is of odd order. 
 Let A^**^ be a skew symmetrical determinant of odd order. Aa.. 
 is a skew symmetrical determinant of even order, and hence 
 
 VA;;r=(-l)'-i(l,2, ..., i-l,i + l, ...,7i) 
 = (i+l, •••, n, 1, 2, .-., i-1). 
 
 Now A^"^ being zero, we have, by 60, 
 
 . A%,= Aa,,A«,,. 
 
 .-.Aa^, =(1 + 1, ...,n,l,2, ..., ^-l) (2) 
 
 (A; + l,...,n, 1,2, ..., ^-1), 
 
 which proves that a bordered skew symmetrical determinant of 
 even order is the product of two Pfaffians : for any minor A^.^ 
 
 * A bordered skew symmetrical determinant is one in which the minor 
 of one of the corner elements is skew symmetrical. 
 
164 
 
 THEORY OF DETERMINANTS. 
 
 of a skew symmetrical determinant is evidently expressible as 
 a bordered skew symmetrical determinant. 
 If 
 
 a,, = -a,,^ we find by (1), 
 
 *«61 
 
 <^12 ^13 ^14 ^15 ^16 
 ^22 ^23 ^24 ^25 ^26 
 
 32 <^33 
 
 a 
 
 34 
 
 *36 
 
 ttfl 
 
 %3 ^54 <^56 <^56 
 
 = -(1,2, 3, 4, 5, 6) (2, 3, 4, 5). 
 
 0,a„ = -a, 
 
 Again, if 
 
 a.. = - a,, 
 
 aa = 
 
 , we find by (2) 
 
 ^«42 
 
 ^21 ^% ^25 ^24 
 
 ttji a23 <^^15 <^14 
 
 «31 0-33 0-35 a34 
 
 (X51 a^ a^ a^ 
 
 (5, 1,2, 3) (3, 4, 5,1), 
 = 0,a^. = -a,. 
 
 as the student can readily verify. 
 
 Circulants. 
 
 132. The resultant of 
 
 f{x) = aia^ + a2^4-«'3 = 0, 
 
 <^W=a^ 
 
 1 = 
 
 0, 
 
 
 Sylvester's method (92) is 
 
 
 
 aj tta «3 
 
 = 
 
 cti 
 
 a2 ttg 
 
 
 Oi ttg ttg 
 
 
 as 
 
 (Xj ttg 
 
 
 Oj a2 (/s 
 
 
 «2 
 
 Oj tti 
 
 
 1 0-10 
 
 
 
 
 10 0-1 
 
 
 
 
 (1) 
 
 (2) 
 
 if. 
 
APPLIC ACTIONS AND SPECIAL FORMS. 
 
 165 
 
 Now tti, ttg, ttg being the three roots of unity, it is evident 
 (94) that 
 
 or, denoting one of the imaginary cube roots of unity by a, 
 the other is a^, and we may write 
 
 R=f{\)f{a)f{a') 
 
 = (tti + a2 + (h) (%a^ + «£« + %) («!« + o.2a^ + ag) , 
 an equation exhibiting the factors of R. 
 
 133. R is evidently a symmetrical determinant formed from 
 the elements ai, a^, a^ in its first row, in such a way that the 
 last element in every row is the first element in every succeed- 
 ing row, and the other elements are written in order. Such a 
 determinant is called a Circulant.* The intimate connection 
 of the Circulant of the third order with the cube roots of unity 
 was shown in the last article. We shall now prove that, in 
 general, the circulant of the nth order, 
 
 0= C(aia2 •••»„) = 
 
 ai 
 
 as 
 
 ttg • 
 
 •• ««-l 
 
 a„ 
 
 «« 
 
 ai 
 
 Cfa • 
 
 •• an-2 
 
 ««-i 
 
 a„_ 
 
 iCtn 
 
 ai . 
 
 •• a»-3 
 
 a«-2 
 
 ... 
 
 ... 
 
 ... . 
 
 ., ... 
 
 ... 
 
 a. 
 
 a, 
 
 05 • 
 
 .. a, 
 
 ^2 
 
 ag 
 
 ag 
 
 ^4 . 
 
 " «n 
 
 «1 
 
 is the product of all factors of the form 
 
 a„a,"-i + a„_iaf"-2 + a„_2a,"-« H h Cfga,^ + aga, + ai = /(aO, 
 
 in which a^ is one of the nth roots of unity, and i accordingly 
 takes successively all the values 1, 2, ••• w. In symbols, we 
 are to show 
 
 C (ai a2 ttg • • • a,„) = 11 (a„ a^"-^ + a^-i^-^ -\ f- ^2 «* + ^i) 
 
 = /(ai)/(<'2)/("a) -/K)- 
 Write another determinant of the ?ith order 
 
 The Circulant is of frequent occurrence in the Theory of numbers. 
 
166 
 
 THEORY OF DETERMINANTS. 
 
 1 ttj ai a^ 
 1 a2 ai ai 
 1 ag ai ai 
 
 1 a„ ai ai 
 
 ttg 
 
 n-l 
 
 Multiplying by rows. 
 
 CA = 
 
 /(ai) /(as) 
 
 ai/(«i) a2/(a2) 
 aiVCtti) ^if{o.2) 
 
 /(a„-0 /(a„) 
 
 a«-i/(a„_i) a„f{ai) 
 
 aL/(««-i) a^/Ca.) 
 
 a::l/(a«-0 a:-V(an) 
 
 arv(«i) arvca^) 
 
 Factoring this product, 
 
 CA=/(aO/(a2)-/(a„)A. 
 .-. O =/(a0/(a2)-/(a„) 
 
 = II (anap^ + a^_ia"-2 H h a2ai + «i) 
 
 For an illustration, x y 
 
 y X 
 
 y X 
 
 y X 
 
 y X 
 
 = (x + ai2/) (x + a22/) (x + a32/) (a? 4- a,y) (x + a,y) 
 
 = {^ + y)fx-h\ 
 
 V5-1 , V10 + 2V5 
 
 + 
 
 V 
 
 -10 
 
 = oy' + 2/•^ 
 
 as was evident from the beginning. 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 167 
 
 134. The circulant of the fourth order 
 
 C = 
 
 a. 
 
 ai 02 ^3 
 
 (X^ Ctj M'2 ^3 
 
 CI2 Cl^ 
 
 «! 
 
 can be expressed as a circulant of the second order, as follows. 
 We have 
 
 -0 = 
 
 aj 
 
 -02 
 
 «3 
 
 -a. 
 
 = 
 
 «i 
 
 ^4 
 
 % 
 
 ^2 
 
 as 
 
 -a^ 
 
 «1 
 
 -a. 
 
 
 «3 
 
 «2 
 
 «! 
 
 a4 
 
 a. 
 
 -«! 
 
 «2 
 
 -a.. 
 
 
 a2 
 
 ai 
 
 0^4 
 
 «3 
 
 ^2 
 
 -«3 
 
 a^ 
 
 -ai 
 
 
 ^4 
 
 as 
 
 as 
 
 ai 
 
 The first of these determinants is obtained by interchanging 
 the second and third rows, and multiplying by ( — 1)^; the 
 second is obtained from the first by reversing the order of the 
 rows, and then reversing the order of the columns. 
 
 Multiplying them together, 
 
 ai2_2a2a4+«3^ 
 
 2a^a^-ai-a^ 
 
 
 
 2asai—a^—ai 
 
 ai—2a^a.2+a^ 
 
 
 
 
 
 ^a^a^—a^—a} 
 
 «2^+«/-2aia3 
 
 
 
 ai-la^a^-^a^ 
 
 'la-n^—ai—a^ 
 
 Whence expressing (7^ as the product of two minors, and 
 extracting the square root, 
 
 = 
 
 ai ai — a4 a2 + «3 ^3 — «2 ^4 
 
 «3 «1 — «2 «2 + Cll Ots — «4 «4 
 
 ttgai — a2«2 + «i«3 — Ct4«4 
 
 aitti — a4a2 + ^gas — a2a4 
 
 as was to be shown. 
 
 The method employed in this special case is equalh^ appli- 
 cable to show that, in general, a circulant of order 2n can he 
 expressed as a circulant of the nth order. 
 
 We add the following proof, however, which is based upon 
 the fundamental property of circulants. 
 
1G8 
 
 THEORY OF DETERMINANTS. 
 
 We have to show that 
 
 G: 
 
 (hn-l ^2» ^1 
 
 ^2n-l ^2n 
 Oj2n-2 ^2n-l 
 • • • ^2«-2 
 
 6. 
 
 62 
 
 
 6»- 
 
 -1 
 
 &n 
 
 6» 
 
 61 
 
 
 K. 
 
 -2 
 
 6n-l 
 
 &»- 
 
 ,J„ 
 
 
 K 
 
 -3 
 
 &n-2 
 
 ... 
 
 ... 
 
 
 ... 
 
 
 ... 
 
 h 
 
 6. 
 
 
 61 
 
 
 6. 
 
 h 
 
 63 
 
 
 6„ 
 
 
 61 
 
 ttg a^ Ois '" «1 «2 
 
 0^2 % <^4 ••• ^2» C^i 
 
 where 
 
 hi = ttitti — a2«a2 + <*2«-i<^3 — f- ... — a2a2n 
 
 62 = astti — agag + o^^a^ — \. ... — a^a^n 
 
 hi = a^ai — a^a2 + «3^3 — +•••— «6«2n 
 
 ^k — ^2*-l^l — <^2ft-2^2 "I" <*2;t-3^3 ^" *** ^2;t ^2n« 
 
 The first determinant 
 
 (1) 
 
 (2) 
 
 Now for every 2 nth root a of unity there is one —a. Hence 
 (2) may be written 
 
 C = if (&na>-' + &.-ia>-4 + ... 4- 63a/ 4- &2a/ + h) • (3) 
 
 i=i 
 
 If ± ttj, ± ttg, ± ttg, ±04, ••, ±a„, 
 
 are the 2 nth roots of unity, it is evident that 
 
 222 2 
 
 ttj , 02 , as , •••, a,i , 
 
 are the nth roots of unity. Hence the second member of (3) 
 equals the second determinant of (1), which establishes the 
 theorem. 
 
 For example, 
 
 = 
 
 a h c d 
 
 = 
 
 E F 
 
 d a b c 
 
 
 F E 
 
 c d a h 
 
 
 b c d a 
 
 
 
APPLICATIONS AND SPECIAL FOBMS. 
 
 169 
 
 in which 
 
 /. C={a' + c'-2bdy-(2ac-b^-d'y. 
 
 Centro-syininetric Determinants. 
 
 135. If we suppose a determinant to be symmetrical with 
 respect to the centre of the square (centro-symmetric*), we 
 have, if the determinant is of order 2w, 
 
 A = 
 
 «!! 
 
 ai2 
 
 
 
 «ln-l 
 
 «ln 
 
 K 
 
 &I2 • 
 
 •• ^n-1 hn 
 
 «21 
 
 ^22 
 
 
 
 «2«-l 
 
 «£« 
 
 &21 
 
 K • 
 
 •• hn~l hn 
 
 ... 
 
 ... 
 
 
 
 ... 
 
 ... 
 
 ... 
 
 ... 
 
 
 
 am 
 
 a«2 
 
 
 
 ««n-l 
 
 Ctnn 
 
 &nl 
 
 &«2 • 
 
 •• Kn-l Kn 
 
 Kn 
 
 Kn- 
 
 -1 
 
 
 bn2 
 
 K, 
 
 Clnn 
 
 «n«-l • 
 
 • ««2 ««1 
 
 ... 
 
 ... 
 
 
 
 ... 
 
 ... 
 
 ... 
 
 ... 
 
 
 
 h^ 
 
 hn-\ 
 
 
 &22 
 
 hi 
 
 «2n 
 
 a2n-l • 
 
 • a22 «21 
 
 &ln 
 
 hn^ 
 
 -1 
 
 
 &12 
 
 6ii 
 
 ttln 
 
 «ln-l • 
 
 • Cli2 «11 
 
 We will transform A as follows : add the last column to the 
 first, the (2n — l)th to the second, and so on, finally adding 
 the (n4-l)th to the nth. Afterward subtract the first row 
 from the last, the second from the (2n — l)th, and so on, 
 finally subtracting the nth. from the (n -f- l)th. Then 
 
 A = 
 
 an 
 
 + &!» 
 
 
 dm 
 
 + ^1 
 
 «21 
 
 + &2« 
 
 
 «2h 
 
 + 621 
 
 Ofnl 
 
 -\-Kn 
 
 
 a«. 
 
 + Ki 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 bn 
 hi 
 
 bn 
 
 K 
 
 hn 
 
 «21 
 «11 
 
 hn 
 K 
 
 * It may be shown that the product of any two determinants of the nth 
 order is expressible as a centro-symmetric determinant of the 2 nth order. 
 
170 THEORY OF DETERMINANTS. 
 
 Hence 
 
 A = «!! + ^m «12 + &1«-1 ••• <hn-l + ^12 ^m + &11 
 
 0^21 + hn ^22 + &2n-l *•• «2n-l + &22 C'in + K 
 
 ttnl 4- Kn Cln2+Kn-1 '•' ^mi-l + &n2 ««« 4" &„1 
 
 Cfnn — ^»1 <^nn-l ~~ ^n2 * * * <^n2 — ^nn-1 ^nl — ^nn 
 
 ^2n — ^21 <^2w-l — ^22 * * * <^22 — 02„_i (X21 — 02„ 
 
 If A is of order 2 n H- 1 , we write 
 
 dll Qq2 
 
 ^21 ^22 
 
 h h 
 
 K K-1 
 
 «1« ^<^1 ^U 
 
 Ct2n "'2 ^21 
 
 Oil A*i «!„ 
 
 v-1 
 
 6ln 
 
 &2«-l 
 
 62. 
 
 ... 
 
 ... 
 
 6„»-i 
 
 Kn 
 
 ?. 
 
 k 
 
 0„2 
 
 0,1 
 
 ai2 
 
 a,! 
 
 By making just the same transformations as before, wc 
 find 
 
 «]1 + &1« «12 4- &1„-1 
 
 ^21 + &2n «22 + &2»-l 
 
 ««1 + &«n «n2 + Kn-1 
 
 21, 2k 
 
 ^nn — ^nl ^nn-1 — "n2 
 
 Ogn— 621 a2n-l— &22 
 
 «ln — &11 <hn-i. — K 
 
 <^2n H" ^21 "^2 
 
 2L r 
 
 ttsi — h^ 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 171 
 
 Collecting results, we have : a centro -symmetric determinant 
 equals the product of two determinaiits each of the nth order, 
 if the order of the symmetric determinant is 2n ; if the order 
 of the symmetric determinant is 2 n + 1 , the factors are of order 
 n and n-\-l respectively. 
 
 For an illustration we expand the following determinant : 
 
 a 
 
 h 
 
 c 
 
 d 
 
 e 
 
 f 
 
 9 
 
 h 
 
 h 
 
 a 
 
 d 
 
 c 
 
 f 
 
 e 
 
 h 
 
 9 
 
 c 
 
 d 
 
 a 
 
 b 
 
 9 
 
 h 
 
 e 
 
 f 
 
 d 
 
 c 
 
 b 
 
 a 
 
 h 
 
 9 
 
 f 
 
 e 
 
 e 
 
 f 
 
 9 
 
 h 
 
 a 
 
 b 
 
 c 
 
 d 
 
 f 
 
 e 
 
 h 
 
 9 
 
 b 
 
 a 
 
 d 
 
 c 
 
 9 
 
 h 
 
 e 
 
 f 
 
 c 
 
 d 
 
 a 
 
 b 
 
 h 
 
 9 
 
 f 
 
 e 
 
 d 
 
 c 
 
 b 
 
 a 
 
 a+h b + g c+f d-\-e 
 
 b-\-9 «+^^ ^ + e c-j-f 
 
 c+f d + e a + h b+g 
 
 d-he c+f b + g a+h 
 
 a — h b — g c—f d — e 
 
 b — g a — h d — e c—f 
 
 c—f d — e a — h b — g 
 
 d—e c—f b—g a—h 
 
 a+h+d+e &4-gr+c+/ 
 b-\-g+c+f a + ^+d+e 
 
 a-^h—d — e b-\-g — c —f 
 b-\-g — c—f a + h — d—e 
 
 a — h-\-d—e b — g-\-c—f 
 b — g + c—f a — h-\-d—e 
 
 a — h—d-\-e b — g — c+f 
 b —g — c-\-f a — h—d-\-e 
 
 Continuants. 
 136. Consider the three simultaneous equations 
 
 (a) 3 iCi — iCg ~ ^ ) 
 
 (b) iCi-f-4a;2— ^3 = 0/' 
 
 (c) X2-\-5xq = ) 
 
172 THEORY OF DETERMINANTS. 
 
 From (a), 
 
 ,(s-|)-, 
 
 X, = 
 
 3^^ 
 
 Xi 
 
 From (p), 
 
 re, = 
 
 From (c), 
 
 X2 
 
 ^__1. 
 
 3 + 
 
 4-^« 
 
 oJa 
 
 aji = 
 
 34- 
 
 4 + 
 
 The value of Xi is thus expressed as a continued fraction. 
 If we solve for Xi by 69, we find 
 
 1 
 
 -1 
 
 
 
 -H 
 
 
 
 4 
 
 -1 
 
 
 
 
 1 
 
 5 
 
 
 3 
 
 -1 
 
 
 
 1 
 
 4 
 
 -1 
 
 
 
 1 
 
 5 
 
 We see then that a continued fraction may be expressed as 
 the quotient of two determinants. 
 
 We shall now proceed to the application of determinants to 
 continued fractions in general. 
 
 137. From the simultaneous equations 
 
 ' (1) aiXi — X2 =ai 
 
 (2) a2Xi + a2X2 = Xs 
 
 (3) asX2-^a^X2 = Xi 
 
 (»-i) 
 («) 
 
 
APPLICATIONS AND SPECIAL FOEMS. 173 
 
 we obtain from (1) 
 
 X, = 
 
 Xo 
 «! 
 
 Substituting in this the values of 
 
 2 3 t 
 
 — J 7 
 
 Xi X2 
 
 as obtained from (2), (3), ••• (n — 1), (n), we have 
 
 Xx = — i — 
 ttj + a2 
 
 «2 + «3 
 
 
 «n-l + a„ 
 
 
 The value of iCi is seen to be expressible as a continued 
 fraction. If we stop at the nth quotient, and thus take the 
 nth convergent for the value of cCi, then x^^i and all the suc- 
 ceeding x's must be conceived to vanish. In that case Xi is 
 the continued fraction. 
 
 jr=^^ 02 ttg a„_i a„ 
 
 The consecutive convergents to F will be denoted by 
 Pi A P^ 
 
 p 
 
 The determinant expression for — ^ is now found b}" making 
 
 Qn 
 
 a„_i = in equations I., and solving for x^ by 69. We find 
 
174 
 
 THEORY OF DETERMINANTS. 
 
 X, = 
 
 aj 
 
 -1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 a^ ■ 
 
 -1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 as 
 
 as 
 
 -1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 a* 
 
 a^ 
 
 -1 
 
 
 
 
 
 
 
 
 ... 
 
 ... 
 
 ... 
 
 ... 
 
 ... 
 
 
 ... 
 
 ... 
 
 ... 
 
 
 
 
 
 
 
 
 
 
 
 
 ttn-l 
 
 On-X 
 
 -1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 a« 
 
 ^n 
 
 ttl 
 
 -1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ttg 
 
 tt2 
 
 -1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ag 
 
 % 
 
 -1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 a«-i 
 
 ttn-l 
 
 -1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 «n 
 
 which is the determinant expression sought, and hence is 
 
 Looking at numerator and denominator of this convergent, 
 we see that 
 
 ^ tti 
 
 and thus 
 
 dai 
 
 «! 
 
 dQn 
 
 _ da, , Tp d(\osQ„) 
 
 138. A determinant having the form of Q„ in the preceding 
 article is called a continuant ; i.e., a continuant is a determi- 
 nant in which the elements outside of the principal diagonal 
 and the two adjacent minor diagonals are all zeros, and one of 
 these minor diagonals has each of its elements — 1 . 
 
 Since 
 
 ttl 
 
 -1 
 
 
 
 
 
 .. 
 
 
 
 
 
 a2 
 
 a^ 
 
 -1 
 
 
 
 .. 
 
 
 
 
 
 
 
 ttg 
 
 as 
 
 -1 
 
 .. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 •* ttn-l 
 
 «n~l 
 
 -1 
 
 
 
 
 
 
 
 
 
 .. 
 
 «n 
 
 On 
 
APPLICATIONS AND SPECIAL FOEMS. 
 
 175 
 
 «1 
 
 ag 
 
 
 
 . 
 
 .. 
 
 
 
 
 
 -1 
 
 a2 
 
 as 
 
 
 
 .. 
 
 
 
 
 
 
 
 -1 
 
 as 
 
 0-4 
 
 .. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ... _1 
 
 ttn-l 
 
 a« 
 
 
 
 
 
 
 
 
 
 ... 
 
 -1 
 
 a« 
 
 it is immaterial on which side of the principal diagonal we 
 write that minor diagonal whose elements are — 1 . Also we 
 may write 
 
 ai 
 
 -1 
 
 
 
 
 
 
 . 
 
 
 
 
 
 02 
 
 (h 
 
 -1 
 
 
 
 
 .. 
 
 
 
 
 
 
 
 ag 
 
 03 
 
 -1 
 
 
 .. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 •• a«-i 
 
 «n-l 
 
 " 
 
 
 
 
 
 
 
 
 
 
 .. 
 
 ttn 
 
 ttn 
 
 «! 
 
 1 
 
 
 
 
 
 . 
 
 . 
 
 
 
 
 
 — ag 
 
 a^ 
 
 1 
 
 
 
 . 
 
 . 
 
 
 
 
 
 
 
 — ttg 
 
 ttg 
 
 1 
 
 . 
 
 . 
 
 
 
 
 
 
 
 
 
 
 
 
 
 . 
 
 • — a«-i 
 
 ttn-l 
 
 1 
 
 
 
 
 
 
 
 
 
 .. 
 
 . 
 
 — a« 
 
 a„ 
 
 We shall employ the following notation : 
 \aia2as"'a„J 
 
 Thus 
 
 tti a2 Og aj 
 
 (h 
 
 -1 
 
 
 
 
 
 «2 
 
 0^2 
 
 -1 
 
 
 
 
 
 as 
 
 «3 
 
 — 
 
 a4 ^4 
 
 Pn 
 
 Returning to ^f » we may now write 
 
 Qn 
 
176 
 
 THEOKY OF DETERMINANTS. 
 
 aj "3 "^ - «« ) 
 Qn / a2 og ... a„ \ 
 
 or, the nth convergent to a continued fraction F is expressible as 
 the quotient of two continuants multiplied by the first numerator 
 ofF. 
 
 139. Expanding P„ in terms of the elements of the last row, 
 we find , . 
 
 \(h as ... a^J 
 
 = a„ai 
 
 as —1 
 
 + a„ai 
 
 "3 
 
 «3 
 
 -1 
 
 
 
 
 
 "4 
 
 «4 
 
 — 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 as 
 
 -1 
 
 
 
 
 
 "3 
 
 «3 
 
 -1 
 
 
 
 
 
 a4 
 
 a. 
 
 — 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^»-l "'n-1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 "-n-S "-n-l 
 
 a„_2 a„_2 
 
 = ana/ "« "^ - «-^ )-\-a^aJ «« «^ - ^-^ ^ 
 Vag ag ... a^_iy \a2 a^ •.• a„_2y 
 
 = «.A-i + a,P,_2. 
 
 Similarly, 
 
 Q^= f a, a, ... a„ ) = aj "^ "« - ^-^ ^ 
 \ai a2 ... a„J \ai ag ... a„_i/ 
 
 -honf "^ "^ '" """-^ ^ 
 
 = «n Qn-1 + an Q»-2. 
 
 (^) 
 
 (5) 
 
APPLICATIONS AND SPECIAL FORMS. 177 
 
 140. It is to be observed that the equations {A) and (B), 
 besides establishing the law of formation of the consecutive 
 convergents to F, give the expansion of a continuant of order 
 n in terms of continuants of lower orders. Thus, by (B), 
 
 / 02 ag a4 \ / ag ttg \ /^ "^ ^ 
 
 \«1 «2/ ^ ^ ^ V«l <^2/ 
 
 = aia2<^3<^4 + aztt^a^ + aiaga^ + 010204 + 0204. 
 
 141. Equation (B) is, in fact, a special case of the more 
 general theorem 
 
 / ttg ttg 04 ••• a„ \ _ / 02 Og •.. "r W a^+2 ••• ttn \ 
 \a^ ..- Or-l/ V^r+2 ••• a„y 
 
 This is easily proved by writing out the continuant of the 
 first member in full, and expanding by Laplace's Theorem 
 (55) in terms of the minors formed from the first r rows and 
 their complementaries. 
 
 We may also use {B) to obtain another expansion of Q„. 
 Thus 
 
 f 02 «3 ••• ttn \ _ f «3 a4 ••• «n \ / ^4 "s *•• «« ^ 
 \a^ 02 ••• On/ ~ "^ Va2 «3 ••• ««/ "^ \«3 «n/ 
 
 as the student may easily verify by expanding the first member 
 in terms of the elements of the first column. 
 
 142. It will afford the student an excellent exercise to take 
 the quotient 
 
 / Og Og ... O,^ N 
 
 \a^ 02 ... a, J 
 
178 THEORY OF DETERMINANTS. 
 
 and, with the help of (C) of the preceding article, transform 
 it into the continued fraction 
 
 Ol «2 «3 ^n 
 
 143. 57 or 62 established the theorem 
 
 dai^dap, dUi^ da^, da^,,, dat^ 
 Let A=q^ = ( "^ '" ^» Y 
 
 and let i = A; = 1, p = e = n. 
 
 Then c^^A ^ f a, ... a„_i A = 5-i^ 
 
 da^ida^^ V«2 «3 ••• ««-!/ "'i ' 
 
 where P„_i has the meaning assigned to it from the beginning. 
 ^=,( ^ '" <^ \ = Pn, 
 
 Also dA 
 
 da^ 
 
 Similarly, 
 
 — — =a2a3 ..• a„; =(_l)«-i; 
 
 dain da„i ^ ^ 
 
 and dA 
 
 da„. 
 
 . / 0.2 as ••• ttn \ f ^S "4 ••• ttn-l \ 
 
 "\a1a2 ••• a^jKa^a^ ••• a„_iy 
 or Q„P«-i - PnQn-1 = (-I)'*aia2a3...a,. 
 
APPLICATIONS AND SPECIAL FOKMS. 
 
 179 
 
 144. With the help of determinants we may now show 
 that 
 
 Oi a2 0-3 
 
 a*-i 
 
 h+i 
 
 +a*-i +«* +a*+i +«„ 
 
 equals 
 
 
 JCtti 
 
 ajtti 
 
 O-k+l 
 
 as follows. The first fraction 
 
 F: 
 
 / ttg a4 ... a„ \ 
 
 tti Va2 «3 ••• «n/ 
 / 02 ttg ... a„ \ 
 
 +a« 
 
 This quotient is easily shown to be equal to 
 
 
 ai 
 
 -1 
 
 
 
 
 
 ... 
 
 
 
 
 
 
 
 
 
 ... 
 
 
 
 
 ^2 
 
 -1 
 
 
 
 ... 
 
 
 
 
 
 
 
 
 
 ... 
 
 
 
 
 "3 
 
 ag 
 
 -1 
 
 ... 
 
 
 
 
 
 
 
 
 
 ... 
 
 X 
 
 
 
 
 
 
 
 
 
 ... a;fc_l 
 
 a*-i 
 
 O-k 
 
 
 
 
 
 ... 
 
 
 
 
 
 
 
 
 
 
 ... 
 
 -1 
 
 Ctk 
 
 -1 
 
 
 
 ... 
 
 
 
 
 
 
 
 
 
 
 ... 
 
 
 
 O-k+l 
 
 «*+l 
 
 -1 
 
 ... 
 
 
 
 
 
 
 
 
 
 
 ... 
 
 
 
 
 
 
 
 
 
 ••• an «« 
 
 
 «! 
 
 -1 
 
 
 
 
 
 ... 
 
 
 
 
 
 
 
 
 
 ... 
 
 
 «2 
 
 ^2 
 
 -1 
 
 
 
 ... 
 
 
 
 
 
 
 
 
 
 ..0 
 
 
 
 
 ag 
 
 a^ 
 
 -1 
 
 ... 
 
 
 
 
 
 
 
 
 
 ..0 
 
 X 
 
 
 
 
 
 
 
 
 
 ... aj^^i 
 
 «*--l 
 
 a* 
 
 
 
 
 
 ..0 
 
 
 
 
 
 
 
 
 
 
 ... 
 
 -1 
 
 a. 
 
 -1 
 
 
 
 ..0 
 
 
 
 
 
 
 
 
 
 
 ... 
 
 
 
 a^+i 
 
 «*+l 
 
 -1 
 
 ..0 
 
 
 
 
 
 
 
 
 
 
 ... 
 
 
 
 
 
 
 
 
 
 .. a„ a. 
 
180 THEORY OF DETERMINANTS. 
 
 But this quotient obviously equals 
 
 a, -1 
 
 
 
 . 
 
 . 
 
 
 
 
 
 
 
 . 
 
 .. 
 
 
 
 a^ 
 
 -1 
 
 . 
 
 .. 
 
 
 
 
 
 
 
 . 
 
 .. 
 
 
 
 03 
 
 % 
 
 -1 . 
 
 .. 
 
 
 
 
 
 
 
 . 
 
 .. 
 
 
 
 
 
 ... 
 
 ... . 
 
 
 
 ... 
 
 ... 
 
 ... 
 
 ... . 
 
 
 
 ... 
 
 
 
 
 
 . 
 
 •• ^O.k-1 
 
 a;a,_i 
 
 Xaj, 
 
 
 
 . 
 
 .. 
 
 
 
 
 
 
 
 . 
 
 " 
 
 -1 
 
 «* 
 
 -1 
 
 . 
 
 .. 
 
 
 
 
 
 
 
 . 
 
 .. 
 
 
 
 a*+i 
 
 «*+l 
 
 -1 . 
 
 .. 
 
 
 
 
 
 
 
 . 
 
 .. 
 
 
 
 
 
 
 
 . 
 
 ' Cln 
 
 a« 
 
 «! — 1 
 
 
 
 . 
 
 .. 
 
 
 
 
 
 
 
 . 
 
 .. 
 
 
 
 a.2 a^ 
 
 -1 
 
 . 
 
 .. 
 
 
 
 
 
 
 
 . 
 
 .. 
 
 
 
 ag 
 
 «3 
 
 -1 . 
 
 .. 
 
 
 
 
 
 
 
 . 
 
 .. 
 
 
 
 
 
 
 
 . 
 
 •• ^k-l 
 
 a^a*-i 
 
 Xaj, 
 
 
 
 . 
 
 .. 
 
 
 
 
 
 
 
 . 
 
 .. 
 
 -1 
 
 au 
 
 -I 
 
 . 
 
 .. 
 
 
 
 
 
 
 
 . 
 
 . 
 
 
 
 "ft+i 
 
 «*+l 
 
 -1 . 
 
 .. 
 
 0. 
 
 
 
 
 
 . 
 
 . 
 
 
 
 
 
 
 
 . 
 
 . a^ 
 
 a» 
 
 This last quotient is 
 
 the continued fraction 
 
 
 
 
 F' = — "^ 
 as was to be shown. 
 
 Xaj,_i Xa^ ttj+i 
 
 +a;a,_i +«* +«&+! 
 
 145. In a certain investigation it becomes necessary to show 
 that the denominators Di and D^ of the convergents to the 
 fractions 
 
 b b 
 
 b 
 
 and 
 
 b b 
 
 b 
 
 tti +a2 
 
 +a„ 
 
 
 an +a«-i 
 
 +a^ 
 
 are equal. 
 
APPLICATIONS 
 
 AND SPECIAL FOKMS. 
 
 We have 
 
 
 
 
 
 
 
 A = 
 
 ttj 
 
 -1 
 
 
 
 
 
 .. 
 
 
 
 
 
 > 
 
 
 b 
 
 ag 
 
 -1 
 
 
 
 .. 
 
 
 
 
 
 
 
 
 
 6 
 
 as 
 
 -1 
 
 .. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 .. b 
 
 a„_i 
 
 -1 
 
 
 A 
 
 
 
 
 
 
 
 ' 
 
 .. 
 
 b 
 
 «n 
 
 
 A = 
 
 «« 
 
 -1 
 
 
 
 
 
 .. 
 
 
 
 
 
 
 
 6 
 
 «n-l 
 
 -1 
 
 
 
 .. 
 
 
 
 
 
 
 
 
 
 6 
 
 «n^2 
 
 -1 
 
 .. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 .. b 
 
 a. 
 
 -1 
 
 
 
 
 
 
 
 
 
 
 
 .. 
 
 b 
 
 ai 
 
 
 181 
 
 By reversing the order of the columns in Da, and also the 
 order of the rows, and afterward making the rows the columns 
 in order, the original determinant is unchanged either in sign 
 or magnitude. But by these transformations D^ is changed 
 to Di. Whence Dg = A> as was to be shown. 
 
 146. The quotient 
 
 62 C3I 
 
 Itti 62 C3I 
 can be expressed as a continued fraction, as follows 
 
 I&2 C3I 
 
 \ai 62 C3I 
 
 1 61 Ci 
 
 
 62 C2 
 
 
 63 Cs 
 
 
 ai bi Ci 
 
 
 ^2 &2 C2 
 
 
 a^ &3 Ca 
 
 
 1 
 
 ttg ^2 I ^1^2! 
 «3 ^3 I ^1^3! 
 
 0^2 ^2 1^1^21 
 % ^3 I ^1^3! 
 
182 
 
 THEORY OF DETERMINANTS. 
 
 63 
 
 l«2&3l ^2 I^Cal 
 
 h I&1C3I 
 
 = 
 
 &3 
 
 Ia253l 
 
 
 
 
 -63I61C2I 
 
 
 
 -1 
 
 !&lC3l 
 
 63 I61C3I 
 
 
 
 -1 
 
 &2 
 
 -I&1C2I 
 
 
 -1 
 
 l^C3l 
 
 h _ 
 
 53I&1C2I 
 
 
 6IC3I 
 
 \ 
 
 This process is equally applicable to show that, in general, 
 the quotient of two determinants — is expressible as a con- 
 tinned fraction, provided only that 
 
 d^ 
 
 dA. 
 
 Ai=--^, or Ai = — ^, or Aj = 
 dan ^^in 
 
 147. The continued fraction 
 
 «! +«2 +«3 
 
 da„ 
 
 or A, = 
 
 dAg 
 
 H-^n 
 
 is evidently equal to 
 
 / a2 ttg ••• a^ \ 
 Vai 02 ••• ct»/ 
 
 For 
 
 a, -1 ... 
 a2 -1 ... 
 ag as -1 ... 
 
 
 
 "n «n 
 
 + 
 
 F,= 
 
 m 
 
 -1 
 
 
 
 ... 
 
 
 
 
 
 Cll 
 
 -1 
 
 .. 
 
 
 
 
 
 a2 
 
 a, -1 
 
 .. 
 
 
 
 
 
 
 
 ttg ag 
 
 ... 
 
 
 
 
 
 
 
 
 
 •• On 
 
 ftn 
 
 ,«1 
 
 Og ttg ... a„ \ 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 183 
 
 But the first determinant in the numerator may be written 
 
 0-10 
 
 ai tti —1 
 
 tta ag — 1 
 
 tto a. 
 
 
 
 -1 
 
 
 
 
 
 «! 
 
 
 
 -1 
 
 
 
 
 
 
 
 a2 
 
 — 
 
 
 
 
 
 ttg 
 
 «3 
 
 
 
 
 
 
 
 
 
 ... 
 
 = 
 
 ... 
 
 
 ... 
 
 
 ... 
 
 
 ••• On «n 
 
 
 
 
 
 
 ... 
 
 
 
 ... 
 
 
 
 ... 
 
 
 
 ... 
 
 
 
 ••• ttn 
 
 Ctn 
 
 whence the desired result is at once obtained by substituting 
 in the numerator of the value of F^, and adding the deter- 
 minants. 
 
 148. We may, with the help of the preceding article, express 
 the value of the periodic continued fraction 
 
 m + 
 
 h ^2 &3 
 
 +02 +«i 
 
 h 
 
 as the quotient of two determinants. (The * marks the re- 
 curring period.) 
 
 If we put X for the continued fraction, we have 
 
 X = m ■+■ 
 
 h ^2 ^3 
 «i +02 +»3 
 
 h 
 
 4-a2 +<^i m+a; 
 
 Then, by 147, 
 
 / bi 62 ••• h h h \ 
 
 \m % 0^2 ••• ttg «! (m-{-x) J 
 
 62 h 
 
 / 62 h " 
 
 \ai Og 0^3 ••• «2 ^1 (^ 
 
 clearing of fractions, and expanding, 
 
 / 62 h "' 63 h &A ^^f h h 
 
 _/ 61 62 ••• h h \ X ( ^^ ^2 
 ~"\m Oi tta ••• ^2 «! m / "^ \m «! a^ 
 
 ^2 (h) 
 
184 
 
 THEORY OF DETERMINANTS. 
 
 But the first term of the first member equals the second 
 term of the second member of this equation. 
 
 x = ± 
 
 ( '' 
 
 
 63 62 ^1 
 
 
 ^ h h '" h h 
 
 «! ^2 «3 ••• Cts «2 «1 
 
 149. Let us now consider the ascending continued fraction 
 
 as 4- .* a« 
 
 — ^ « ^ tti +a2 +a3 
 
 
 ^- = ». + "^ 
 
 
 enote the convergents to F^ by 
 
 
 P Pi i>2 Pn 
 _, — , — , ... — f 
 
 ^ qi q2 qn 
 
 
 and let us obtain the determinant expression for the nth con- 
 vergent. 
 
 We have evidently 
 
 p^ is determined from the following equations, which the 
 student can easily deduce : 
 
 — O'zPt-^Pz 
 
 = a3 
 
 — ttn-a P«-8 + Pn-t = an-2 
 
 — an-lP«-2 + Pn -1 = a^-l 
 
 a«P»-i+i>» =0^. 
 
APPLICA.TIONS AND SPECIAL FORMS. 
 
 185 
 
 From these equations 
 
 i>n = 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ttl 
 
 -^2 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 02 
 
 
 
 -^3 
 
 1 
 
 
 
 
 
 
 
 
 
 ttg 
 
 ... 
 
 ... 
 
 ... 
 
 
 ... 
 
 
 ... 
 
 ... 
 
 ... 
 
 
 
 
 
 
 
 
 a„ 
 
 ~2 
 
 1 
 
 
 
 an-2 
 
 
 
 
 
 
 
 
 
 
 - 
 
 -«n-l 
 
 1 
 
 O-n-1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 <^n 
 
 a« 
 
 ai 
 
 -1 
 
 
 
 . 
 
 
 
 
 
 
 
 
 
 ^2 
 
 a^ 
 
 -1 
 
 . 
 
 
 
 
 
 
 
 
 
 "3 
 
 
 
 «3 
 
 -1 . 
 
 
 
 
 
 
 
 
 
 a«-2 
 
 
 
 
 
 . 
 
 • «'»-2 
 
 -1 
 
 
 
 
 a„-i 
 
 
 
 
 
 . 
 
 
 
 
 ttn-l 
 
 -1 
 
 
 ttn 
 
 
 
 
 
 . 
 
 
 
 
 
 
 ttn 
 
 
 
 ttl 
 
 -1 
 
 
 
 . 
 
 .. 
 
 
 
 0.2 
 
 ^2 
 
 -1 
 
 . 
 
 .. 
 
 
 
 as 
 
 
 
 «3 
 
 -1 . 
 
 .. 
 
 
 
 ... 
 
 ... 
 
 ... 
 
 ... . 
 
 
 
 ... 
 
 a«-i 
 
 
 
 
 
 . 
 
 •• «n-l 
 
 -1 
 
 O-n 
 
 
 
 
 
 . 
 
 .. 
 
 dn 
 
 ai 
 
 -1 
 
 
 
 . 
 
 . 
 
 
 
 
 
 a^ 
 
 -1 
 
 . 
 
 ■• 
 
 
 
 
 
 
 
 ^3 
 
 -1 . 
 
 . 
 
 
 
 ... 
 
 ... 
 
 ... 
 
 ... . 
 
 
 
 ... 
 
 
 
 
 
 
 
 . 
 
 •• ttn-l 
 
 -1 
 
 
 
 
 
 
 
 . 
 
 . 
 
 a„ 
 
 Pn 
 
 150. The numerator and denominator of - can be trans- 
 
 formed into continuants, and thus the fraction F^ can be 
 transformed into a descending continued fraction, as follows : 
 
186 
 
 THEORY OF DETERMINANTS. 
 
 Multiply the last row of p„ by (v-u and subti-act from it the 
 (n — l)th row multiplied by a„ ; then multiply the (n— l)th 
 row by a„_2, and subtract from it the (w — 2)th row multiplied 
 by a„_i ; and so on. Then 
 
 «! 
 
 -1 
 
 
 
 
 
 
 
 agai+ag 
 
 — ai 
 
 
 
 
 
 — agoia ^so^j+as 
 
 — ou 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Pn = 
 
 — a„_2a„_i an_ia„_2-|-a„_i — a„_2 
 
 a„_ia„_2a„_3 
 
 a^a^ai 
 
 Similarly, 
 
 gn = 
 
 a, -1 
 
 
 
 ... 
 
 
 
 
 
 
 
 — aittg asttj + aa 
 
 — ttl 
 
 ... 
 
 
 
 
 
 
 
 — tta^s 
 
 «3a2-f-a3 
 
 — ttg ... 
 
 
 
 
 
 . 
 
 ... 
 
 ... 
 
 
 
 ... 
 
 ... 
 
 ... 
 
 
 
 
 
 ... 
 
 a«-i 
 
 a«-ia»-2 
 + an-i 
 
 — an-2 
 
 
 
 0* 
 
 ... 
 
 
 
 -««-!«« 
 
 
 an-ia»-2an-3 
 
 a2ai 
 
 Whence, by 144, 
 
 Ol 
 
 a^a^ 
 
 Ct2«l«3 
 
 Ol --a2ai+a2 — asOg+ag 
 
 5'« 
 
 • 
 
 an-2an-3a^-l 
 
 a„_ia„_2an 
 
 — «n-ia„-2+an_i 
 
 — ««««-! +an 
 
 the descending fraction sought. 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 187 
 
 Alternants, 
 151. Consider the determinant 
 
 A = 1 «! Of/ ••• ( 
 
 1 ttg a^ 
 1 a„ a„2 
 
 as 
 
 and the product 
 
 P = (a2-ai)(a3-ai)(a4-ai) ... K-ai) 
 
 X (as— «2)(a4— «2) ••• («n— 02) 
 X (a^-as) ••• (a^-ag) 
 
 X (a^-a„_i) 
 
 of the - (71— 1) differences of the n different quantities involved 
 
 in A. This product is called the difference product of the n 
 quantities ai, ag, ••• a„, and for it the notation ^*(ai, ag, ag, ••• a„) 
 has been adopted. 
 
 The reader will remember that the square of the difference 
 product was denoted by ^(ai, as, ••• a„), and thus the difference 
 product itself is very appropriately designated by C*(ai, ag, ••• a„). 
 
 We shall now show that 
 
 A =P = CK«15 «2, --J ttn)' 
 
 (1) 
 
 If in A we put a^ = a„^ A vanishes ; hence A is divisible by 
 each factor of P, and hence by P. Again, A and P are each 
 
 n 
 polynomials of degree - (n — 1), and therefore 
 
 A = \CH«15 «2, «3? ••-,«„), 
 
 where A. is a factor independent of ai, ag? •** <*»*• From the 
 special case 
 
188 
 
 THEORY OF DETERMINANTS. 
 
 
 (a^-a,) (ccg-^i) (a3-«2)> 
 
 we see that X= 1, and thus the truth of (1) is established.* 
 
 152. A of the preceding article is evidently an alternating 
 function ; for the interchange of a^ and a„ amounts to an 
 interchange of two rows in the determinant, and hence changes 
 its sign. A is accordingly an Alternant. In general, an alter- 
 nant is a determinant in which each element of the first row is 
 a function of £Ci, the corresponding elements of the second row 
 the same functions of x^, and so on. Thus 
 
 /»K)] 
 
 A = 
 
 fiixO Mx,) ... M^,) 
 Mx,) L{x^ - f.{x,) 
 
 Mx,d Mx:) ... Ux^) 
 
 ■^AlMx,), Ux,) 
 
 is an alternant. 
 
 153. We can easily show that 
 
 A = 
 
 1 fx{x,) Mx,) ... u,{x,) 
 
 1 fl(X2) A{X2) '•' fn-l(x,) 
 
 = H^(Xi, X, 
 
 
 1 /iW /sW - fn 
 
 -l{Xn) 
 
 
 • x^), 
 
 where fr(x) is a function of the rth degree in x, and A. is the 
 product of the coefficients of the terms of highest degree in 
 the several functions. For subtracting the first column mul- 
 tiplied by the proper number from the second, we reduce the 
 elements of the second column to piX^, piX2, p^x.^, ... PiX,^. 
 Then subtracting the sum of the first and second columns, each 
 multiplied by the proper number, from the third column, the 
 elements of this column become ^2^/? ^2^2'*) ••• P2^n^. Pro- 
 ceeding in this way, we see that finally 
 
 A = \tj{xi, ^2, ••• a;„). 
 
 See also examples 6 and 7, page 37. 
 
ArPLICATIONS AND SPECIAL FORMS. 
 
 189 
 
 where ^ = Pi • i>2 • • • Pn* 
 
 For an example, putting 
 
 we have 
 A 
 
 1 /iW f2M ... f^_,(x{) 
 
 1 Mx,) Mx,) ... f,,_,(x,) 
 
 1- /l(^n) /.(^n) - /n-l(^«) 
 
 (71-1)! (n-2)! ... 2!' 
 
 154. Every alternant whose elements are rational integral 
 functions of Xi, X2, ••• »„, is divisible by t}{xi^ iCg? ^31 ••• ^n)» 
 and the quotient is a symmetric function of the variables. 
 For the alternant vanishes if Xt = Xj,, and hence is divisible by 
 Xi — x^, and thus by CH^n ^2? ••• ^n). '^^^ quotient must be ^ 
 symmetric function, for the interchange of Xi and Xj, changes 
 the sign of both dividend and divisor ; therefore the sign of the 
 quotient remains unchanged upon the interchange of two of 
 the variables, and is accordingly a symmetric function. We 
 shall now actually perform the division just considered. Alter- 
 nants whose functions are powers of the variables are called 
 simple alternants^ and are of frequent occurrence. We proceed 
 first to the discussion of simple alternants. 
 
 155. The quotient 
 
 1 X, 
 
 ... arr'^i' 
 
 1 X, 
 
 "'xr'x2' 
 
 
 
 
 1 x^ 
 
 -'X^--'X^^ 
 
 A(Xj\x2, Xs^"'Xl_-i,X„^) 
 
 -f- i {X,, X,, ... X^) = ^^^-_____^ 
 
 may be developed as follows : 
 
 Expand the dividend A in terms of the elements of the last 
 column, and we obtain 
 
190 
 
 THEORY OF DETERMINANTS. 
 
 da^i' die/ dx/ dxj 
 
 Now, it is evident that each of the minors in this expansion 
 is a difference product. 
 Thus 
 
 dA 
 dxj 
 
 ^(-1) 
 
 -L 3/j flJj 
 
 ,n-2 
 
 1 «,_i a^_i ••• x^i! 
 1 OJ,.^! a^^i ••• x^^i 
 
 1 a7„ £C„ 
 
 icr 
 
 Substituting in (1) the values of the minors as found from 
 (2), and dividing both members of (1) by l^{xi, iCg, ••• a;„), we 
 have a series of terms, of which 
 
 (-!)»+" a;/ 
 
 lx^-X,){x„_-^-X,) ... {Xr+i — Xr) (X,-X^_i) •" (x, — X2){X,-Xi) 
 
 is the type. Thus we find 
 
 C-{Xi, X2, ..., x„) 
 r=i {x^-Xr)(x^^i~x,) ... (a;,+i-a;,)(a;,-a;,_i) ... {x^-x,) ' 
 
 or 
 
 a^i' 
 
 4.... + 
 
 
 (X,- 
 
 -a;,) (»,- 
 
 -w„ 
 
 -.) 
 
 ••(^1 
 
 -X,) 
 
 
 + 
 
 
 
 
 x,^ 
 
 
 
 
 (».- 
 
 -a!„)(X2- 
 
 -a;. 
 
 -.)• 
 
 ..(0^- 
 
 -X3) (asj- 
 
 -a;,) 
 
 
 
 
 
 
 ^L. 
 
 
 
 (aJ„-2-««)(a;„_2-a?n-i)(»«-2-»n-3) ••• (a;«-2-aa) 
 
APPLICATIONS AND SPECIAL FOKMS. 
 
 191 
 
 + ^ 
 
 For an illustration, we have 
 
 1 2 16 
 
 -f- 
 
 1 3 81 
 
 
 1 5 625 
 
 
 2 
 
 4 
 
 3 
 
 9 
 
 5 
 
 25 
 
 16 
 
 + 
 
 81 
 
 625 
 
 = 69. 
 
 (2-5) (2-3) (3-5) (3-2) (5-3) (5-2) 
 
 156. With the help of the preceding article we may reduce 
 the quotient 
 
 ^i(a;i, ajg, ••• «„) 
 
 to the sum of two similar and simpler quotients, as follows : 
 Since 
 
 A{X^, X}, Xi, ... <:f, V) - ^n^i^X-, ^2, 
 
 K'-l. ^n'-') 
 
 1 x^ x^ 
 
 1 X2 xi 
 
 
 1 X^_i af„_2 •«. flJ^-l ^n-l(^n-l — ^n) 
 
 1 x^ 
 
 XI 
 
 
 
 (1) 
 
 we have, after dividing both members of (1) by l^{xi, x^, ••• x^) 
 in accordance with 155, and striking out the factor common 
 to numerator and denominator of each term in the second 
 member, 
 
 A{x,\ x}, xj, ..., x^-l x,^) x,A{x,\ x,\ x,\ ... <:f, x^') 
 
192 
 
 THEOBY OF DETERJVaNANTS. 
 
 q-l 
 
 ^i 
 
 ?-l 
 
 + ... + ^ 
 
 K-i-aJ»-2) {x^-\-Xn-z) ••• (a^n-i-a^i) 
 
 But the sum in the second member of this equation is, 
 by 155, 
 
 1 Xi Xi 
 
 J. Xo Xo 
 
 x^-'^ x^-'^ 
 
 1 ^n-1 Xn-\ ••• 
 
 Transposing, we have 
 
 /v.n-3 
 
 ^H-1 
 
 -^ lK^i'>^2'> •••ja;„_i). 
 
 ^H^i' ^2, ••• a;„) ^K^i5 ^2, ••• x^) 
 
 + 
 
 A.{Xi , a72 , i^3 , 
 
 a^^Il) 
 
 ^n-3 ^?-r 
 
 which is the desired reduction. For example, 
 
 \ X a? 
 
 
 1 a; «2 
 
 
 
 1 y f 
 
 
 1 y f 
 
 
 1 a.-2 
 
 1 2 ^ 
 
 
 1 ^ 2^ 
 
 1 
 
 1 2/2 
 
 ^i(a;, 2/, 2!) 4Ha^» ^5 ^) ^H^» 2/) 
 
 = a; + y + 2 = 2a;. 
 
 I X x'^ 
 
 
 1 X a^ 
 
 
 
 1 2/ 2/* 
 
 
 1 2/ f 
 
 
 1 a!« 
 
 1 ;2^ 
 
 
 1 « 2? 
 
 , 
 
 1 2/» 
 
 The student may show 
 
 = '^a^ + 'Z^y-^-xyz, 
 
 1 
 
 X 
 
 a^ 
 
 1 
 
 y 
 
 f 
 
 1 
 
 z 
 
 z" 
 
 ^H^5 2/5 2;) 
 
APPLICATIONS AND SPECIAL FORMS. 193 
 
 -^x^ + %a^y + 5a^2/2 _^ ^^^^^ 
 
 1 
 
 X x' 
 
 1 
 1 
 
 y f 
 
 z z' 
 
 4* (^, y, z) 
 
 157. Since every term of CK^i' ^2^ ••• ^n) contains a permu- 
 tation of all the powers of the variables from 1 to 71 — 1 , each 
 
 term is of the -(n — l)th degree. Similarl}-, every term of 
 
 A(x,\ x,\ xi, ..., <:f, 0^,0 is of degree iliZliliZLlA) + g. 
 
 Hence every term of 
 
 f;^^ A{x,\x,\xi, '".xlzlx^) , 
 
 i\x,, X2, '", X,) 
 
 is of the (q — n -\- l)th. degree, as is illustrated in the exam- 
 ples of the preceding article. We shall now show that every 
 possible term of the (g — n4-l)th degree in the variables is 
 found in Q, and that every such term is positive. That is to 
 say, the quotient Q is the complete symmetric function of degree 
 (g- 71 4-1) of a^i, X2, •••, x^. 
 Such a term of Q is 
 
 1 =i Xi X2 x^ '•' X n_2 ar„_i x^. 
 
 By successively applying 156, we develop Q so that the 
 terms containing ic„, aj^a^^i, x^x^^_ix\,^2i etc., are at once 
 distinguished. In the first place, 
 
 ^ ^«,a;2^a^3^'••,<:i,a^r-l) . x.,A(x,\x2\x,\ •^-.xZzlxlzl) 
 t}{xi, X2, ..., x^__{) ^i(a;i, X2, •.., a;„_i) 
 
 ^ aiA(x,',X2\xi,...,xl-lxt_\) ^ ^ xr''A(x^,X2\xir';x:izixZz]) 
 
 The second term, 
 
 X^Ai^Xi , X2 , X^^ ' • » , a?;i,„2^ ^ra 1 ) 
 
 = «1, 
 
194 THEORY OF DETERMINANTS. 
 
 contains the first power of x^ ; hence we must look for T in 
 Qi. Applying 156 to Qi, as before, we have 
 
 
 ^-1^(^1 ^ ^2 ? ^3 ^ "'^ ^n-3? ^n-2) I 
 t (^li ^21 •*•? ^n-2) 
 
 a^n:r'^(«^l^ ^2\ «^3', -, ^n-l) j^^ 
 
 -'} 
 
 4 (a?!, a;2i * * * 5 ^n 2) 
 
 In this expansion we must look for T in the third term 
 
 % 
 
 ^n^n-\-^\^\ ^ ^2 ? ^3 ^ •••? ^n- 3? ^n '2) __ Q 
 t (^1? ^*2? '"j ^n-i) 
 
 Q2 may be expanded as before ; continuing in this way, we 
 finally obtain the term 
 
 X aP x^ ...T« ^W^2') . 
 
 ^^{X^X2) 
 
 for the coefficient of a7„«l_ia;^„-2 ••• a^s^ contains only a^i and a?2, 
 and is of the third degree. Upon performing the division, and 
 multiplying, one of the terms is T. Since T is any term, the 
 proposition is established. 
 
 Employing the notation H^ for the complete symmetric 
 function of the rth degree, we may write the result of the 
 present article 
 
 A(x,\x,\x,\....xlzlx,^,) _ jj (^ X ... x\ 
 
 or simply Hq^^+i* 
 
 For illustrations the student may refer to the examples in 
 the preceding article. 
 
 « With this notation, Hq = 1, ZT., = 0. 
 
APPLICATIONS AND SPECIAL FOKMS. 
 
 195 
 
 Again, 
 
 1 X x^ x^ 
 
 1 y y^ f 
 
 1 z ^ ^ 
 
 1 t f f 
 
 = iJ^ = -^x* + 20^2/ + Ix^y^ + ^xyzt. 
 
 l'-{x, y, z, t) 
 
 158. From the two preceding articles we have at once 
 
 H^{X^,X^, "•,X^) = X^H,_.l(Xi,X2, '••,X^)-\-Hr(Xi,X.,, •••,i»«_i). 
 
 (1) 
 
 Whence we readily obtain 
 
 Hr.i{x„x.>, '•■,x^+i) = x^^^H,_ci{x^,x^,"',x^^^)-^H,_^{x^,X2, ..., x^y, 
 Substituting in (1), 
 
 -"r(^l? ^2» "'-i ^n) = ^nL-"r-l(^l5 ^2> '**? ^n+l) 
 
 - x„+i^,_2(a;i, a^a, •••, Xn+i)~\+H,{x^, x^, ..., a;„_i). (2) 
 Similarly, 
 
 XZy(i»l, a;2» •••> ^n— l^«+l) = ^n+lL-"r-l(^l? ^25 '"^ ^n+l) 
 
 From (2) and (3), 
 
 Hri^l-, a^2, ••• ^n) —^ri^l, X^, '" X^_iX^+i) 
 
 = (^n- Xn+l)Hr-l(Xi, X2, ' - • iK„+i) . (4) 
 
 159. If any alternant whose elements are powers (simple 
 alternant) be divided by the difference product of its variables, 
 the result is expressible as a determinant whose elements are 
 complete symmetric functions of the variables. That is to say, 
 
 Ajx'^, xi, ... 0^::) 
 
 
 
 
 -fftt-n+l H^-n+l 
 
 H^- 
 
 -n+l 
 
196 
 
 THEORY OF DETERMINANTS. 
 
 This may be proved as follows. For brevity we employ 
 determinants of the third order, but the method applies, of 
 course, to determinants of any order.* In the alternant 
 
 
 Xi 
 
 x^ x^ 
 
 (1) 
 
 subtract the first row from each of the other two, then remove 
 the factors {x^ — Xi) , (x2 — Xi) . Afterward subtract the second 
 row from the third, and remove the factor ccg — a^g? employing 
 equation (4) , 158. The result is 
 
 A{ xi, Xj, x^) _ 
 
 ll^{X,) ' ll^(X,) ffy(X,) 
 
 H^_.2{X^,X2,^5) H^-2{X^,X2,X^) IIy-2{Xi,X2,X^) 
 
 The determinant on the right we now transform as follows. 
 Add the second row, multiplied by ajg, to the first, employing 
 equation (1), 158, and obtain the determinant 
 
 IIa{Xi, X2) H^i^lt i»2) Hy{X^^ Xo) 
 
 IIa-2{Xi, X2, Xs) Hp-2(Xi, X2, ajg) Hy-2{X^, X2, x-i) 
 
 Now add the third row, multiplied by ajg, to the second, 
 again employing (1) of 158; finally, add the second row, 
 multiplied by iCg, to the first. 
 
 We then obtain 
 
 ^i(a;i, X2, x^) 
 
 Ha(x^,X2,Xs) B'p(Xi,X2,Xs) Hy(x^,X2,Xs) 
 
 Ha-l(Xi,X2,Xs) n^-l{X^,X2,X^) Hy-l{Xi,X2,X^) 
 Ha-2(Xi,X2,Xs) H^.2{Xi,X2,X^) Hy-2{Xi, X2, X^) 
 
 as was to be shown. For an example, 
 
 * The mode of proof here given is due to Mr. 0. H. Mitchell, American 
 Journal of Mathematics, Vol. IV., page 344. 
 
APPLICATIONS AND SPECIAL EOHMS. 
 
 197 
 
 a 
 
 a' 
 
 a' 
 
 = abc 
 
 h 
 
 b' 
 
 b' 
 
 
 c 
 
 c' 
 
 (^ 
 
 
 = abc^^{a,b,c) 
 
 = abc^^{a,b,c) 
 
 1 a^ a* 
 1 b^ b' 
 1 c" c^ 
 
 HQ{a,b,c) IIs(a,b,c) H^{a,b,c) 
 
 -Hi ^2 
 
 = abGt^{a, b, c) | Sa^ + ^ab ^a" + %a-b + %abc 
 
 = a6c^i(«» ^ c) I - 2a5 - Sa^^ - 2 2a6c 
 1 2a 2a' + 2a6 
 
 =^abc^^{a,bj c) 
 
 — 2a6 2a6c 
 2a — 2a6 
 
 160. Form the product 
 
 an 
 
 ai2 • 
 
 • «1« 
 
 X 
 
 xr' 
 
 ^-» . 
 
 • a^i 
 
 1 
 
 «21 
 
 ^22 • 
 
 • «2n 
 
 
 a,r' 
 
 
 • X2 
 
 1 
 
 ... 
 
 
 
 . ... 
 
 
 ... 
 
 ... . 
 
 . ... 
 
 .. 
 
 a„i 
 
 «n2 •• 
 _i 
 
 
 . /I... 
 
 
 i»,"-2 . 
 
 • ^n 
 
 1 
 
 changing the columns of the first determinant into rows before 
 multiplying. If we put 
 
 f,{x) = a^.x''-^ + a^.x''-"- H \- a^_^,x + a„„ 
 
 we find 
 
 p=(_l)l(-^>|a,J^K^1.^2,-,aJn) 
 
 /l(»2) f2(X2) ••• fn{X2) 
 /l(^») /2(a'n) ••• /«(a;„) 
 
193 
 
 If now 
 we must have 
 
 laiJ = 
 m-1 
 
 / 
 
 TRBOBT OF DETERMLNANTS. 
 
 Mx.)^^x.-y;)'-\ 
 
 1 (:T)i-y^^ Wy-y^y [vy-y^y 
 
 
 1 (7')^-^") {^ty-y^^y (Vy-y^y - (-^»)"-^ 
 
 where 
 
 But this last determinant evidently equals 
 
 where K is the product of all the binomial coefficients of order 
 w — 1. We have, accordingly, 
 
 (aJi-2/i)"-' (a^i-2/2)""' - (aJi-2/n)""' 
 («2-2/i)""' (^^2-2/2)""' ••• (a?2-2/n)"~' 
 
 (x^-y^y-' {x^-y^y-^ - K-2/n)"-' 
 
 = /r^*(^l> «'2, a?3, •••, a^n) ^*(2/l» 2/25 2/3, •••, 2^n) • 
 
 If now x^ — y^y we have C(^i> ^2? ^s> •••^^n) in the form of a 
 determinant. 
 
 161. Siippose now that aj, ag, •••,a„ are the roots of an 
 equation 
 
 f(x) = 0. (1) 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 199 
 
 Then ^^(ai, a^, ag, •••, a„) is the product of the differences of 
 the roots of (1). Square this determinant, obtaining 
 
 ^ (ttj, as, • • • , an) = 
 
 1 + 1 +- + 1. 
 
 «! +CL2 H f-a„ 
 
 a/ +ai +...+a,f 
 
 «! +a2 H f-an 
 
 ai^ + a^^ + .-.+a^f 
 
 tti 
 
 n-l 
 
 af +a2'* + 
 
 + a 
 
 + a„" 
 
 + a2"-i+...+a,'*-i 
 
 -"+' + 02"+' + 
 
 + a 
 
 n+l 
 
 a,'--'+ai--' + 
 
 H-a„^' 
 
 So 
 
 Sl 
 
 S2 
 
 
 Sn-1 
 
 Si 
 
 S2 
 
 S3 
 
 
 s» 
 
 S2 
 
 S3 
 
 s. 
 
 
 ^n+l 
 
 ... 
 
 ... 
 
 ... 
 
 
 ... 
 
 s«- 
 
 iSn 
 
 Sn+1 
 
 
 S2»-2 
 
 where, as usual, 
 
 S^ = ai*- + a2'*+ ••• +a,r. 
 
 162. The preceding article gives us an expression for the 
 square of the differences of the roots in terms of s^. We can 
 also readily obtain an expression for the sum of the squares 
 of the differences in terms of s< as follows. 
 
 We have 
 
 by 58. 
 
 1 1 
 
 P y 
 
 Sl 
 
 Si So 
 
 = 2(a-^y 
 
 163. We shall conclude our discussion of alternants with a 
 theorem on the reduction of alternating functions to alternants.* 
 
 * " Reduction of Alternating Functions to Alternants," Wm. Woolsey 
 Johnson, American Journal of Mathematics, Vol. VII., page 345. 
 
200 
 
 THEORY OF DETERMINANTS. 
 
 (1) 
 
 Any function of the form 
 
 4>i{a, bed ••• I) 4>2(iii bed ••• I) ••• <^„(a, bed ••• I) 
 <f>i{b, aed '" I) </)2(6, acd ••• Z) ••• <j>n(b, aed "- 1) 
 
 r 
 <f>i{l, abc ••• k) cf>2{l, abc ••• A:) ••• ^^(Z, abc ••• k) 
 
 is evidently an alternating function of a,b,c, "• I, if 
 
 <^(ci, 6ccZ ••• Z) 
 
 denotes a function of the n quantities a^b,c,'"l, which is 
 symmetrical with respect to all the quantities except a. If 
 each element of this determinant contains only the leading 
 letter, (1) becomes 
 
 Mb) 
 
 M<^) 
 Mb) 
 
 /3(«) 
 
 Mb) 
 
 M') Ml) Ml) 
 
 Mo) 
 Mb) 
 
 Ml) 
 
 (2) 
 
 an alternant, which we represent, as usual, by its principal 
 term, 
 
 [/l(«),/2W,/3(c),-/n(0]. (3) 
 
 Now, if the principal term of (1) can be separated into parts 
 of the form (3), then the given alternating function (1) is 
 equal to the sum of the alternants represented by these partial 
 terms. This is proved as follows. Since an interchange of 
 two rows of (1) is equivalent to an interchange of the corre- 
 sponding letters, any term of (1) can be obtained from the 
 principal term by a suitable transposition of the letters, and, 
 similarly, the corresponding term in each of the alternants may 
 be derived from its principal term by the same transposition 
 of the letters ; hence every term in the expansion of (1) is 
 equal to the sum of the corresponding terms in the expansion 
 of the alternants. 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 201 
 
 Accordingly, if a determinant of the form (1) is expressed, 
 as usual, by writing its principal term in (), with commas 
 between the elements, we may erase the commas, and treat the 
 expression within the ( ) as an ordinary algebraic quantit}'. 
 
 Thus, 
 
 = A{bcd, 1, c, d^) =^« 6, r, d^) = t,^{a, b,c, d). 
 
 bed 
 
 1 
 
 a 
 
 a' 
 
 cda 
 
 1 
 
 b 
 
 W 
 
 dab 
 
 1 
 
 c 
 
 (? 
 
 abc 
 
 1 
 
 d 
 
 d' 
 
 Again, 
 
 1 b^ + (? a' + bc 
 1 c2-fa2 b^ + ca 
 1 a' + b^ c' + ab 
 
 =^(l,c2 + a2, c2 + a6) 
 
 A{a\ b\ c') -\-A(a\ b\ c') +A{a, b, c') +A{a\ 6, c«) 
 - A {a\ 6, c^) = _ (a -h 6 + c) C* (a, b,c). ' 
 
 Functional Determinants. 
 
 164. Consider the following n functions of the n independent 
 variables iCi, X21 ••• x^. 
 
 2/i=/i(a?i,a;2, "',x^) 
 
 (1) 
 
 These functions will be independent if for every set of values 
 of yi,y2,"'yn equations (1) determine one or more sets of 
 values of a^i, ajg, ••• a;„, so that these latter variables can in their 
 turn be considered as functions of the n independent variables 
 
 yi,y2,"'yn' 
 
202 THEORY OF DETERMINANTS. 
 
 Differentiating equations (1), we have 
 
 dy, = ^'dx, + pdx2 + .-. + ^dx^ 
 
 hXi 
 hXi 
 
 dy2=^'dx,-\-^dx2 + 
 
 Sx^ 
 
 SX2 
 
 Sx^ 
 
 dy, 
 
 ^^Adx^+^^dx2 + ••• +^d.x„ 
 
 hxi 
 
 Sxo 
 
 6x„ 
 
 (2) 
 
 Regarding equations (2) as a system of equations for deter- 
 mining dxi, dx2, •" dXn, the determinant of this system 
 
 S?/i 
 
 S«2 
 
 S.Vi 
 
 
 ••,2/.) 
 
 %2 
 
 8x, 
 
 
 %2 
 
 
 
 8& 
 
 8& 
 
 %, 
 
 
 
 8a;, 
 
 8x, 
 
 ■ 8*„ 
 
 
 
 is called the Jacobian of the given functions yi^ 2/2? * * • Vn' 
 Or, in other words, the Jacobian of a set of n functions, each 
 of n variables, is the determinant |A:i„l, in which the element 
 kpg is the first derivative of the pth function with respect to the 
 ^th variable. Thus, given 
 
 yi = az^ -\-2 hzt + ci^, 2/2 = ^i^!^ + 2 hiZt + Cif. 
 
 The Jacobian 
 
 
 
 az + 6i hz -\- ct 
 a^z + &i^ h^z 4- Ci? 
 
 _4 1 
 
 
 4 
 
 1 
 
 = 4 
 
 ^2 _2« 22 
 
 . 
 
 "^, 
 
 2/1 hz -\-ct c 
 
 
 a 6 c 
 
 
 ^' 
 
 Vi h,z H 
 
 - Ci« Ci 
 
 
 a, « 
 
 '1 Cj 
 
 
APPLICA.TIONS AND SPECIAL FORMS. 
 
 203 
 
 165. If the functions 2/1,2/2, ••• 2/n ^^^ iiot independent, but 
 are connected by a relation 
 
 <^ (2/15 2/2, •••2/n) = 0, 
 
 the Jacobian vanishes. 
 
 From (3) we have, by differentiating, 
 
 (3) 
 
 %l ^Xl 82/2 ^^1 
 
 82/1 80^2 82/2 8a;2 
 
 82/n S«^i 
 
 H- ^ . ^" = 
 S2/« 80^2 
 
 (4) 
 
 §i . ?^i _i_ ^ . 5^2 ^ ... _i_ ^ . ^ = 
 
 82/1 8a;« 82/2 8aj, 8y^ Sx^ 
 
 From their mode of formation, .equations (4) are simulta- 
 neous. Hence the determinant of the system vanishes by 77 ; 
 
 ^ = 0. 
 
 We shall show presentl}^ that if the Jacobian of a set of 
 functions vanishes, the functions are not independent. 
 
 166. The Jacobian of the implicit functions 
 
 Fi (a^i, X2,'" cc„, 2/1, 2/2, ••• 2/n) = ^ 
 F2 (a^i, X2, ••• a;„, 2/1, 2/2? • • • 2/n) = 
 
 (5) 
 
 Fn(Xi, »2, ••• X^, 2/1? 2/2J ••• 2/n) = 
 
 is found as follows. 
 Equations (5) yield 
 
 _SF\^§T\ . S^i_^S5 8^2 ^ ... ^ 55- . %2 
 Sxj, 82/1 Bxj, 82/2 Sx^ Sy^ Bx^ 
 
 (1, A;=l, 2, 
 
 (6) 
 
 n). 
 
204 THEORY OF DETERMINANTS. 
 
 Using equation (6), we find the product of 
 
 to be 
 
 8F, 8F, 
 
 SF^ 
 
 and 
 
 
 %2 
 
 8x, 
 
 Byn 
 
 '" 8x, 
 
 8F2 SF, 
 
 %1 %2 
 
 8F, 
 
 
 8x, 
 
 5^2 
 
 6X2 
 
 8x2 
 
 BF^ 8F„ 
 
 Byi %2 
 
 SF^ 
 
 
 Syi 
 
 8x„ 
 
 %2 
 
 Byn 
 
 '" 8x^ 
 
 (-1)" 
 
 SF, 8F, 
 
 ... 
 
 8F, 
 8x^ 
 
 • 
 
 
 SF^ 8F2 
 8x1 8x2 
 
 ... 
 
 8F2 
 
 
 
 Si^n S^n 
 
 
 BF^ 
 
 
 
 8x^ 
 
 .8x2 
 
 ... 
 
 Sx^ 
 
 
 Whence 
 
 j^ Hyi.y2,'-yn) ^. 8(j^i,j^„-..i^0 . bjf^f^^-'F^) 
 
 8{xi, X2, ••• x^) 8{xi, X2, — x„) ' 8(2/1, 2/2» ••• 2/n) 
 
 (7) 
 
 If in (7) we put n= 1, we get 
 
 8F\_8F\ dyi 
 8x1 8yi dxi 
 
 a well-known formula. 
 
 167. If in equations (5) we consider ajj, ajg? •" x^ as func- 
 tions of yi, 2/2> ••• 2/rn W6 obtain, as above, 
 
 r^y S{F„F2r"F,) ^ 8iF,,F2r"F,) ^ 8(x,,X2,'"X^) ,g. 
 8(2/1,2/2, •••yn) S(a;i,a;2, •••»„) 8(2^1,2^2, ••• 2/«)* 
 
(9) 
 
 APPLICATIONS AND SPECIAL FOBMS. 205 
 
 From (7) and (8), 
 
 8(a;i, x^, ... x„) 8(2/1, 2/2, ... y^) 
 
 168. Again, having given the n -{-p functions, 
 
 2^1 (xi, x^, ... a;,,, 2/1, 2/2, ••• 2/n+p) = 
 F^ix^, %, ••• a?„, yi, 2/2, — 2/«+;,) = 
 
 Fn-^{xi, 02, — a;„, 2/1, ^2, ••• 2/n+i>) = 
 The Jacobian 
 
 J-^ 8(2/1, ^2, — Vn) 
 
 o(aJi, a;2, ••• x^) 
 
 of the first w of these functions is found as follows. Differen- 
 tiating equations (10), we find 
 
 (10) 
 
 8x, Sy, Sx, "^ By, Sx, "^ "^ Sy„+, So;, 
 
 (6) 
 
 
 (* = 
 
 1,2,... 
 
 7i+i>; 
 
 A:=l,2, 
 
 ...n) 
 
 Now multipl}' together 
 
 
 
 
 A = 
 
 8i^i 8i^i 
 82/1 82/2 
 
 8i^i 
 
 S2/n+p 
 
 X 
 
 S2/1 
 
 %2 
 
 8a;i •*' 
 
 S2/n 
 
 8a;i 
 
 
 8F2 8i<T, 
 %i 82/2 
 
 8F2 
 
 
 S2/1 
 
 8^2 
 
 ^2/2 
 8a;2 
 
 S2/n 
 80^2 
 
 
 %! 8^2 
 
 SF^^P 
 ^Vr^p 
 
 
 %1 
 
 8cc„ 
 
 %2 
 
 8a^n *'* 
 
 82/n 
 
 first writing J" as a determinant of order w -j-p? thus : 
 
206 
 
 THEOBY OF DETERMINANTS. 
 
 8^1 
 
 
 8xi 8xi 
 
 BX2 SX2 
 
 8x, 8x^ 
 1 
 
 
 
 Sxi 
 
 SX2 
 
 Calling the product P, we have 
 
 p^(-iy 
 
 8a;i 
 
 SaJi 
 8P„. 
 
 8Pi 8Fi 
 
 8a;„ 
 
 8y„+i 
 
 
 
 s-f;+. 
 
 8-F-n+, 
 
 Sa^i 
 
 8Pi 
 
 n+p 
 
 SP„ 
 
 = (-!)% 
 
 8a;„ 82/„+i 
 
 S(Pi, ^21*** ^^i+p) 
 
 ^Vn+p 
 
 6 (a?!, a;2» • • • ^n» 2/n+l» 2/n+2> • • • Vn+p) 
 
 since, by equation (6) for A;^n, the element Ui^ of P is ? ; 
 
 and for A; > n, 
 
 8^ 
 
 gajfc 
 
 We have, accordingly. 
 
 -=£• 
 
 169. Suppose equations (5) yield upon solution 
 2/1 = </>i(a?i, 3^2, •••,a;„). 
 
 (0 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 207 
 
 Solve (c) for x^, and substitute this value of Xi in the 
 remaining ii — l equations ; then 2/2, 2/3, ••• y^ become functions 
 of ?/i, a?2, ••• ic„. Thus 
 
 2/2= <^2(2/i, 372, •••ajn). W 
 
 Solve (c?) for »2? substitute the result in the remaining n — 2 
 equations ; then 2/3, 2/4, ••• 2/n become functions of 
 
 2/l> ^29 ^39 *••? ^n* 
 
 Thus 2/3 = <^3(.yi, 2/2, ^3» •••, a^«). 
 
 Solve (e) for 0^3, substitute as before ; and so on. 
 We obtain the equations 
 
 2/1 - c}>i(Xi,X2, •••«„) =0 
 
 y2 — 4>2{yuX2,'"X^) =0 
 
 3/3- <^3(2/l9 2/2, 3^3, •••«?„) =0 
 
 2/« - <^n(2/l? 2/2, ••• 2/n-l, ••• a„) = 
 
 By 166, 
 
 j^B(y,, 
 
 2/2, •• 
 
 •^n) 
 
 
 
 = (-1)" 
 
 8{x,,x,,'..y^) 
 
 
 
 hXi 8X2 
 
 ^01 ... Bcfji 
 
 8x, " 8x^ 
 
 
 1 
 
 
 
 
 
 -^2 
 Sx, 
 
 Oc/j2 002 
 
 6x, '" g^ 
 
 
 802 
 
 %1 
 
 1 
 
 
 
 
 
 Hs . S03 
 
 3aJ3 K 
 
 
 803 
 
 S2/1 
 
 803 
 
 82/2 
 
 1 
 
 
 
 ... -% 
 
 
 S0n 
 8^1 
 
 80n 
 %2 
 
 B<f>n 
 82/3 
 
 
 _ 8<^, _ 802 . 
 8i»i 8x2 
 
 803 
 
 8x^ 
 
 
 
 (e) 
 
 (11) 
 
208 THEORY OF DETERMINANTS. 
 
 That is to say, the Jacohian of a set of functions 2/1, 2/2? ••• 2/n» 
 each of n independent variables x^, 0^2, ••• a;„, is expressible as a 
 product of n differential coefficients of the functions <f>i, <^2> * • • S^n* 
 where </)^ is a function of ?/i, 3/0, ••• 2/r-i» ^n ••• ^n- 
 
 170. The result just obtained may be employed to show that 
 if the Jacobian of a set of functions vanishes, the functions are 
 not independent. 
 
 For, if r=5ii.?& ... Hn. 
 
 hxi 8x2 8x^ 
 
 vanishes, some one of the coefficients, say 
 
 where i has one of the values 1, 2, ••• n. But if -^ = 0, <^< 
 does not contain cc^, i.e., * 
 
 Vi = ^iiVi^ 2/2, ••• Vi-i, ^i+u ••• a^n). 
 Also ?/,+i = <^f+i (2/1, 2/2, ••• 2/i5 ^i+u •••«'«). 
 
 From these two equations, 
 
 2/i+i = «Ai+i (2/1^ 2/2? ••• 2/t? »<+2, a^i+s, ••• ^«) ; 
 
 therefore 2/i+i does not contain x^^-^. In the same way we may 
 show that 2/<+2 ^^^^s not contain a;^^.2, and so on. Hence, 
 finally, 
 
 2/«= «Ah (^1^2/2, •••2/«-i); 
 
 or y^ is expressible as a function of the remaining n — 1 func- 
 tions, and hence the given functions are not independent. 
 For example, if the given functions are 
 
 (1) u=x-\-y, (2) v=:x — z, (3) w = qi^ -\-xz — yz — z^, 
 
 1 1 2/ + ^ 
 
 1 x-z 
 
 -1 x-y-2z 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 209 
 
 J evidently vanishes. Accordingly, (1), (2), (3) are not 
 independent. That the given functions are not independent is 
 easily shown directly as follows. We readily obtain 
 
 y z=u — 'V — Z. .'. W = V {u — V), 
 
 as was to be shown. 
 
 171. If the functions y^, y^, ••• y^ are the n partial derivatives 
 
 ic„) , the Jacobian 
 
 V-' /"' •*• -r-' of a function /(a?!, 0^2, 
 hXi bxo bx^ 
 
 •ff(/) = 
 
 
 sy 
 
 ixiSxi 
 
 sy 
 
 ay 
 
 sxi 
 
 sy 
 
 sy 
 
 Sx^Sxi 8x^8x2 
 
 ay 
 
 8xi8Xn 
 
 jy_ 
 
 Sx28x^ 
 
 sy 
 
 8x^' 
 
 is called the Hessian of (x^, x^, • • • x^ . The Hessian is a 
 symmetrical determinant, since 
 
 jy_ = J!/L. 
 
 SXiSxj, Sxj^BXi 
 If the derivatives -^, -— ••• r^^ are connected by an equa- 
 
 OXi 6X2 ox,^ 
 tion, with constant coefficients 
 
 dXi 6X2 Sx^ 
 
 the Hessian must vanish. 
 
 172. Let /i, /2, "• fn be n given functions of the same 
 variable x. Suppose the functions are connected by the linear 
 relation 
 
 tti/i + a-,fo + aj, + ... + a„/^ = 0, (1) 
 
210 
 
 THEORY OF DETERMINANTS. 
 
 in which ai, ag, •••a„ are not functions of x. Differentiating 
 (1) successively n — 1 times, we have 
 
 ai/l" +«2/2" +«3/3" +-«n/u" =0 
 
 Eliminating oti, ag, ••• «« from (1) and (2), we find 
 
 (2) 
 
 /i fi h 
 
 f\ fi fz 
 ■fit fti ftt 
 
 Jl J2 J& 
 
 /'n-1 -fn—l -fn—l 
 
 
 = -D (/!,/«/« •••/,) = ()• 
 
 (3) 
 
 The determinant of (3) has been called the Wronskian of 
 fii f2y •" fn' We see from (3) that if the functions /i,/2, •••/» 
 are connected b}" a linear equation of the form (1), the Wron- 
 skian vanishes. 
 
 173. If we denote the given functions by 2/1? 2/2? ••• 2/n? and 
 the derivatives by 2/11, 2/21? •*• {i'^'-> the second subscript denoting 
 the derivatives), we may write (3) 
 
 Vn 
 
 y2 
 
 Vn 
 
 = D{yi,y2iy3, •••2/n)=o. 
 
 Vln-i 2/2 «-l ••• Vnn-l 
 
 Now y being any function of », we find 
 
 2/"-0(yi5 2/2,2/3, •••2/«)= yiy iyiy)i 
 » (2/22/) 1 
 
 y^ (yny)i 
 
 (yiy)n-i 
 
 (2/22/) n-l 
 (2/n2/)n-l 
 
 (4) 
 
 (5) 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 211 
 
 in which the subscript k of {yiy)k means the kih derivative of 
 {yiy). That is to say, the Wronskian of 2/i2/? 2/22/v ••• 2/n2/ is 
 the product on the left in (5). This is made evident by notic- 
 ing that since 
 
 (2/i2/)i = ViiV + ViV^ {yiy\ = 2/<22/ + ^yav' + ViV", 
 
 etc., where y', y", •••, are the successive derivatives of ?/, the 
 determinant on the right becomes a sum of determinants, of 
 which the first is the product on the left, and all the rest 
 vanish. 
 
 174. We find 
 
 
 
 dD{y^,y2,-',yn) 
 
 Vi 
 
 yn — yin-2 2/ln 
 
 dx 
 
 2/2 
 
 2/21 ••• 2/2n-2 2/2n 
 
 
 Vn 
 
 2/«i — 2/»«-2 2/«« 
 
 (-1) 
 
 for in the sum of determinants which make up the derivative 
 sought, all vanish except the one expressed in equation {A) . 
 
 175. If in 173 we put y = —, the Wronskian on the right in 
 
 (5) reduces to 
 
 2/1 
 
 .2/1/1 
 
 ri/2 
 '2/3' 
 
 V2/1A V2/1/2 
 
 'y2' 
 .2/1 
 
 \yJn-i 
 
 D 
 
 yA fys' 
 .2/1/1 
 
 Now 
 
 '2/2 
 
 i/i~ y? ' Wi" yi ' "* \yJ~ y? 
 
212 THEORY OF DETERMINANTS. 
 
 Then if we put 
 
 D (?/i, 2/2) = 2:2, D (2/1, 2/3) ^z^, '" D (2/1, 2/„) = 2!n1 
 we get 
 
 ^ (2/1, 2/2, ••• 2/n) =-;n^ fe, 23, ••• O- (6) 
 
 2/1 
 
 176. We shall employ the result just obtained to show that 
 if the Wronskian of 2/15 2/2^ •** 2/» vanishes, the functions are 
 connected by a linear equation having constant coefficients. 
 Suppose that 2/1 does not vanish, and since by hypothesis 
 
 D{yi^y2^ •••2/n) = 0, 
 by (6) of the last article we must also have 
 
 yr' 
 
 Therefore, by 172, the n — 1 functions Zg? 2:3, z^ are connected 
 by a linear relation, i.e., 
 
 0.2Z2-^asZs-\ f-an^n=0. (7) 
 
 Dividing (7) by 2/1^, and restoring the values of Z2, z^, ••• 2;,„ 
 
 Integrating (8), we find 
 
 «i2/i + «2y2 + «32/3 H h a„2/n = 0. (9) 
 
 Therefore assuming that if the Wronskian of n — 1 func- 
 tions vanishes, the functions are connected by a linear relation, 
 we have shown that when the Wronskian of n functions van- 
 ishes, the functions are connected by a linear relation. But 
 the assumption is obviously true for two functions, hence the 
 theorem is true universally. 
 
APPLICATIONS AND SPECIAL FORMS. 
 
 213 
 
 Linear Substitution. 
 177. If the n functions (one or more) 
 
 /g = O2I a^2 + <^22 ^2 H + a^n^n 
 
 (1) 
 
 are transformed into functions of 2/i»2/2? '•• Vn ^J the following 
 linear substitutions,,* 
 
 «1 = ^l2/l + &12 2/2 + ••• + Kyn 
 ^2 = &212/1+&222/2 H h &2u2/n 
 
 »n= &«iyi + &n22/2 + •'• + &„„?/« 
 
 (2) 
 
 the determinant I 6i„ I of the system (2) is called the modulus 
 of transformation. If the modulus is unity, the substitution 
 is unimodular. If a^j, iCg, • • • »« are independent, the modulus 
 cannot vanish. 
 
 178. If the functions (1) are transformed by means of (2) 
 into 
 
 /i = mn2/i + mi22/2H (-Wi„?/„ 1 
 
 /a = msi 2/1 + m22 2/2 H f- m,,, y. 
 
 (3) 
 
 frf= l^nlVl + Wln22/2 H h W„,.2/„ 
 
 the determinant of the system (3) , 
 
 Imn W22 ••• m„^l, 
 
 * The learner can understand the importance of linear substitution by 
 noticing that such a substitution is the process involved in transformation 
 of coordinates in Geometry. 
 
214 
 
 THEORY OF DETERMINANTS. 
 
 equals the product of the determinant of the given sjstem (1) 
 by the modulus of transformation. That is to say, 
 
 \m,^\ = I«,J X |6i«I. 
 
 This is proved as follows. The coeflficient of y,^, 
 
 Wl« = ttii 6u + «,2 ^2* H h Clin &«*» 
 
 is found by multiplying equations (2) by ciii? ^f2» ••• «m? respec- 
 tively, and adding by columns. Whence, by 53, we see that 
 
 will 
 
 mi2 ••• min 
 
 = 
 
 «n 
 
 ai2 
 
 77121 
 
 m22 '" rn^n 
 
 
 «21 
 
 ^22 
 
 ... 
 
 
 
 
 ... 
 
 ... 
 
 m,a 
 
 w„2 ••• m„„ 
 
 
 a. 
 
 a«2 
 
 «2h 
 
 ^1 hi ••' hn 
 621 622 ••• hn 
 
 &„i b„o ... 6„„ 
 
 179. If /(iCi, aJa, '" x^) is to be so transformed by the sub- 
 stitution 
 
 i»l = /8„ 2/1 + A2 ^2 + — + /?l„ Vn ] 
 i«2 = /?21 2/1 + /?22 2/2 H h Au 2/n 
 
 ' 
 
 5„= ;8„i^i + ^„22/2 -I (- ;3„„2/„ J 
 
 (a) 
 
 that 
 
 2// + 2/1 + - + ^n' = ^1' + a:/ + - + a^,f, 
 
 the linear substitution is called orthogonal. The coefficients of 
 an orthogonal substitution must satisfy the following condi- 
 tions. 
 A. Since 
 
 yi'+yi-h'-'+Vn' 
 
 = (/3ii2/i+ft2 2/2 + -+A»2/«)'+(Ai2/i+&2/2 4- - +ftn2/n)^ 
 
 + +(/8«l2/l+)8„22/2+-+)8nn2/u)' 
 
 + ••• + 22/12/2 (ftii8i2 + ft,&+ - +i8„i;8„2) 4- -, 
 
APPLICATIONS AND SPECIAL FORMS. 215 
 
 we must have 
 
 X fAf +Af +'" + ^J =1 
 
 lA^A* + AA*-f •••+A.-/5«. = (t, A:=l,2, ... n). 
 
 B. If we wish to return to the original function from the 
 transformed function, we must put 
 
 For from (a) we readily find 
 
 Aia^i + /32i^2H hAu^^n 
 
 = 2/i(Ai Ai + Ai Ai + - Ai AO + 2/2(^2 Ai+&/32i+ - +/5„2 A,) 
 
 + - + ^nCAn A. + An Ai + - + An AO ' 
 
 Now, by I., the coefficient of ?/j = 1, and the other coefficients 
 vanish. 
 
 C. The square of the determinant of the S3'stem (a) 
 (modulus of transformation) is unity. 
 
 For 
 
 Al A2 - A: 
 
 Ai fe - A, 
 
 III. 
 
 = IAJ^=IA«!. 
 
 Al A2 - An 
 \Din\ is a symmetrical determinant by 108 ; since, by I., 
 
 A*=o, A=i, 
 
 the truth of III. is obvious. 
 
 D. Bi„ being the minor of ySf^ in I y8i„ I , we find 
 
 ^.■* = AJA«1- 
 
 For multiplying the equations 
 
 AiA* + - +AiA* = o, 
 
 A*A* + -+A*A*=i, 
 ... ... ... ...J 
 
 A»A* + -+AnA* = o, 
 
216 THEORY OF DETERMINANTS, 
 
 ill order by Ba, Bi,, -•' Bi^, and adding, we have 
 
 But all the coefficients, except the coefficient of ^i^, vanish ; 
 hence 
 
 IV. 5a=AJAnI- 
 
 E. By the preceding condition IV., 
 
 (Ai^a + - + A-u/?*n) I AJ = Bal3,i 4- - + An/3*«. 
 
 The second member of this equation is IjSiJ, or 0, according 
 as i and k are equal or unequal. 
 Whence 
 
 1 Ali^H + A2 ^2 + • • • + An)S,. = 0. 
 
 i^. The following relation holds between the minors of the 
 modulus of the orthogonal substitution. 
 
 VI. 
 
 Pr+l r+1 Pr+l r+2 
 Pr+2 r+1 Pr+2 r+2 
 
 Pn r+1 Pn r+2 
 
 For, by 61, 
 
 
 lAnI X 
 
 Al A2 
 Al ^822 
 
 A 
 
 Ai y8^ - Ar 
 
 J521 i522 ... B,, 
 
 B. 
 
 Brr 
 
 = lAnl 
 
 r--l 
 
 A+l r+1 A+1 r+2 
 A+2 r+1 Pr+2 r+2 
 
 A r+1 A r+2 
 
 An 
 
APPLICATIONS AND SPECIAL FORMS. 
 Now, by ly., 
 
 B.21 B.22 '" B^r 
 
 B^, Bo 
 
 B,, 
 
 = I An l^ X 
 
 Ai fe 
 
 P21 P22 
 
 f^rl A2 
 
 21T 
 
 Whence, equating the second members of these two equa- 
 tions, the relation VI. follows. 
 
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