UC-NRLF $B SM3 bMM ii HniiW P iliillilPlf lllllliliiililiilliiilffl^ ill Hill ■ ^H^wi uiiu HI f WW r [jiilii! ! f! HI ^^^^^Hi^^ i JUjiiiiiiiH ' III ' ■ 'M ii . ^ ■ •,: 1 ■ i 1 t 1 { ! \4\m ill LIBRARY University of California. Class Digitized by the Internet Archive in 2008 with funding from Microsoft Corporation http://www.archive.org/details/determinantsOOweldrich MATHEMATICAL MONOGRAPHS. EDITED BY Mansfield Merriman and Robert S. Woodward. Octavo, Cloth, $i.oo each. No. 1. HISTORY OF MODERN MATHEMATICS. By David Eugene Smith. No. 2. SYNTHETIC PROJECTIVE GEOMETRY. By George Bruce Halsted. N^No. 3. DETERMINANTS. By Laenas Gifford Weld. S^No. 4. HYPERBOLIC FUNCTIONS. By James McMahon. V^No. 5. HARMONIC FUNCTIONS. By William E. Byerly. No. 6. QRASSMANN'S SPACE ANALYSIS. By Edward W. Hyde. No. 7. PROBABILITY AND THEORY OF ERRORS. By Robert S. Woodward, No. 8. VECTOR ANALYSIS AND QUATERNIONS, By Alexander Macfarlanr. No. 9. DIFFERENTIAL EQUATIONS. By William Woolsey Johnson. Vno. 10. THE SOLUTION OF EQUATIONS. By Mansfield Merriman. No. 11. FUNCTIONS OF A COMPLEX VARIABLE. By Thomas S. Fiskb. PUBLISHED BY JOHN WILEY & SONS, NEW YORK. CHAPMAN & HALL, Limited, LONDON. MATHEMATICAL MONOGRAPHS. EDITED BY MANSFIELD MERRIMAN and ROBERT S. WOODWARD. No. 3. DETERMINANTS. BY LAENAS GIFFORD WELD, Professor of Mathematics in the State University of Iowa. FOURTH EDITION. ENLARGED. FIRST THOUSAND. NEW YORK: JOHN WILEY & SONS. London: CHAPMAN & HALL, Limited 1906. Copyright, 1896, BY MANSFIELD MERRIMAN and ROBERT S. WOODWARD UNDER THE TITLE HIGHER MATHEMATICS. First Edition, September, 1896. Second Edition, January, 1898. Third Edition, August, 1900. Fourth Edition, January, 1906. ROBERT DRUMMONP, PRINTER, NEW YORK. EDITORS' PREFACE. The volume called Higher Mathematics, the first edition of which was published in 1896, contained eleven chapters by eleven authors, each chapter being independent of the others, but all supposing the reader to have at least a mathematical training equivalent to that given in classical and engineering colleges. The publication of that volume is now discontinued and the chapters are issued in separate form. In these reissues it will generally be found that the monographs are enlarged by additional articles or appendices which either ampHfy the former presentation or record recent advances. This plan of publication has been arranged in order to meet the demand of teachers and the convenience of classes, but it is also thought that it may prove advantageous to readers in special lines of mathematical Hterature. It is the intention of the publishers and editors to add other monographs to the series from time to time, if the call for the same seems to warrant it. Among the topics which are under consideration are those of elliptic functions, the theory of num- bers, the group theory, the calculus of variations, and non- Euclidean geometry; possibly also monographs on branches of astronomy, mechanics, and mathematical physics may be included. It is the hope of the editors that this form of pubHcation may tend to promote mathematical study and research over a wider field than that which the former volume has occupied. December, 1905. 235463 AUTHOR'S PREFACE. The author of the present volume feels some embarrassment in having already offered to the public a v^ork upon the Theory of Determinants. The apparently general acceptability of this former v^ork, which has now reached its third edition, doubtless led to his being invited by the editors of Higher Mathematics to prepare for them a chapter upon the same subject. This was done without the least thought of its pubhcation as a separate volume. Now that its issue as such, along with the other chapters, is requested by both the pubhshers and the editors of Higher Mathematics, it is but just to the author that the above circum- stances should be understood lest he be suspected of entertaining an unseemly desire to keep himself before the mathematical public by vain repetition. The limitations imposed have permitted the addition of only a few articles to the work as originally pubHshed; principally those treating of linear substitutions, quantics, invariance, co- variance, and functional determinants. Determinants of special forms have not been considered, nor is there the least reference to the apphcation of determinants to geometry. It is hoped, however, that the work may prove useful to the constantly increas- ing number of students who, while not wishing to specialize in mathematics, desire to obtain the comprehensive view of its methods and processes essential to the successful pursuit of the exact sciences in general. Iowa City, Iowa, U. S. A., December, 1905. CONTENTS. Art. I. Introduction Page 7 2. Permutations •. . 8 3. Interchange of Two Elements 9 4. Positive and Negative Permutations 10 5. The Determinant Array 10 6. Determinant as Function of w^ Elements 11 7. Examples of Determinants 12 8. Notations 13 9. Second and Third Orders 14 10. Interchange of Rows and Columns .16 11. Interchange of Two Parallel Lines 17 12. Two Identical Parallel Lines 17 13. Multiplying by a Factor 18 14. A Line of Polynomial Elements 18 15. Composition of Parallel Lines 19 16. Binomial Factors 20 17. Co-factors; Minors 21 18. Development in Terms of Co-factors 23 19. The Zero Formulas 25 20. Cauchy's Method of Development 26 21. Differentiation of Determinants 28 22. Raising the Order 29 23. Lowering the Order 30 24. Solution of Linear Equations 31 — ' 25. Consistence of Linear Systems 32 26. The Matrix 35 27. Homogeneous Linear Systems 35 28. Co-factors in a Zero Determinant , ... 37 29. Sylvester's Method of Elimination 38 30. The Multiplication Theorem 40 31. Product of Two Rectangular Arrays 43 32. Reciprocal Determinants 44 33. Linear Transformations 44 34. Qu antics; Invariants and Covariants 45 '6 " ' ' ' ' ' CONTENTS. Art. 35, The Discriminant ' Page 46 36. Composite Quadrics 47 37. Discriminant of Binary Qu antic an Invariant. . . . . 47 38. The Jacobian .48 39. Jacobian of Indirect Functions 49 40. The Jacobian a Covariant , 50 41. Jacobian of Implicit Functions 50 42. The Hessian 51 Index 55 DETERMINANTS. Art. 1. Introduction. As early as 1693 Leibnitz arrived at some vague notions regarding the functions which we now know as determinants. His researches in this subject, the first account of which is contained in his correspondence with De L'Hospital, resulted simply in the statement of some rather clumsy rules for elimi- nating the unknowns from systems of linear equations, and exerted no influence whatever upon subsequent investigations in the same direction. It was over half a century later, in 1750, that Gabriel Cramer first formulated an intelligible and general definition of the functions, based upon the recognition of the two classes of permutations, as presently to be set forth. Though Cramer failed to recognize, even to the same extent as Leibnitz, the importance of the functions thus defined, the development of the subject from this time on has been almost continuous and often rapid. The name " determinant" is due to Gauss, who, with Vandermonde, Lagrange, Cauchy, Jacobi, and others, ra,nks among the great pioneers in this development. Within recent years the theory of determinants has come into very general use, and has, in the hands of such mathema- ticians as Cayley and Sylvester, led to results of the greatest interest and importance, both through the study of special forms of the functions themselves and through their applica- tions.* * A list of writings on Determinants is given by Muir in Quarterly Journal of Mathematics, 1881, Vol. XVIII, pp. 1 10-149. DETERxMINANTS. Art. 2. Permutations. The various orders in which the elements of a group may be arranged in a row are called their permutations. Any two elements, as a and b, may be arranged in two orders : ab and ba. A third, as ^, may be introduced into each of these two permutations in three ways : before either element, or after both ; thus giving 3x2 = 6 permutations of the three elements. In like manner an additional element may be intro- duced into each of the permutations of i elements in (/+ i) ways: before any one of them, or after all. Hence, in .general, if Pi denote the number of permutations of i ele- ments, Pi^^ = {t+ i)Pi. Now, P3 = 3 X 2 X I = 3 ! ; hence P^ = 4 X 3^=41; and, n being any integer, P„ = n{n — i){n — 2) . . . i = n\. That is, the number of permutations of n elements is n\. For all integral values of n greater than unity, n ! is an •even number. If the elements of any group be represented by the differ- ent letters, a, b, c, . . ., the alphabetical order will be considered as the natural order of the elements. If represented by the same letter with different indices, thus : a^, ^„ ^3, . . . ; or thus : a\ a" ^ a'" , . . ., the natural order of the elements is that in which the indices form a continually increasing series. Any two elements, whether adjacent or not, standing in their natural order in a permutation constitute a permanence ; standing in an order which is the reverse of the natural, an inversion. Thus, in the permutation daecb^ the permanences are de, ae, ab, ac ; the inversions, da, dc, db, ec, eb, cb. The permutations of the elements of a group are divided into two classes, viz.: even or positive permutations, in which the number of inversions is even ; and odd or negative permu- tations, in which the number of inversions is odd. 4i. INTERCHANGE OF TWO ELEMENTS. 9 When the elements are arranged in the natural order the number of inversions is zero — an even number. Thus, the even or positive permutations of the elements Y7,, ^„ ^3 are while the odd or negative permutations are Art. 3. Interchange of Two Elements. It will now be shown that if, in any permutation of the elements of a group, two of the elements be interchanged the class of the permutation will be changed. Let q and j be the elements in question. Then, represent- ing collectively all the elements which precede these two by P, those which fall between them by R, and those which follow by Ty any permutation of the group may be written PqRsT. Of the elements R, supposed to be r in number, let represent h the number of an order higher than q, i " " " '' " lower " q, j " ** " " " lower " s, k " " " " " higher " s. It is evident that no change in the order of the elements qRs ■can affect their relations to the elements of either P or T. Then, passing from the order PqRsT to the order PRqsT changes the number of inversions by {h — i) ; and passing from this to the order PsRqT again changes the number of inversions by (/ — /&) ± i, the * p!"^ [sign being used as q is of | ^9^^^^ I order than s. \ mmus ^ ^ ^ ^ ( higher j The total change in the number of inversions due to the inter- change of the two elements in question is, therefore, h — i +y ^ k ± \. 10 .. DETERMINANTS. But since i =^ r — h and k ^= r —j\ this may be written 2{h+J-r)± I, which is an odd number for all admissible values of //,/, and r^ Hence, the interchange of any two elements in a permutation changes the number of inversions by an odd number, thus changing the class of the permutation. Art. 4. Positive and Negative Permutations. Of all the permutations of the elements of a group, one half are even and one half odd. To prove this, write out all the permutations. Now choose any two of the elements and interchange them in each permu- Itation. The result will be the same set of permutations as {even ) odd \ P^^""*uta- tion of the old set has been converted into an \ p„p,, r one in the new. Hence, in either set, there are as many even permu- tations as odd ; that is, one half are even and one half odd. Prob. I. Classify the following permutations: {\) b c d e a \ (2) iii v i ii iv; (3) kninilj', (4) a" a'a' a^' a'"; (5) l^eyt^ad-, (6) 52413; (7) -^1 -^3 -^0 -^4 ^a -^5 ; (S) F. Tu. M. Th. W.; (9) jxkviX, Prob. 2. Derive the formula for the number of permutations of n elements taken m at a time. (Ans. n\/{7t — m)\.) Prob. 3. How many combinations of m elements arranged in the natural order may be selected from a group of n elements? (Ans. n\/m\(n — m)\.) Prob. 4. Show that o! = i. Art. 5. The Determinant Array. Assume ft^ elements arranged in n vertical ranks or columns, and n horizontal ranks or rows, thus : a„ a^ . . . a («) DETERMINANT AS FUNCTION OF «' ELEMENTS. 11 In this array all the elements in the same column have the same superscript, and those in the same row the same subscript. The columns being arranged in order from left to right, and the rows likewise in order from the top row downward, the position of any element of the array is shown at once by its indices. Thus, a^" is in the third column and the fifth row of the above array. The diagonal passing through the elements a^ ^ ^/', . . . ^^^"^ is called the principal diagonal of the array; that passing through a^^ cin-i" y • • • '^/"^ the secondary diagonal. The posi- tion occupied by the element a/ is designated as the leading position. Art. 6. Determinant as Function of «' Elements. ■^■ The array just considered, inclosed between two vertical bars, thus : a/ a/' . . . ^/«> a„ a„ ... a„' is used in analysis to represent a certain function of its «' ele- ments called their determinant.* This function may be defined as follows : Write down the product of the elements on the principal diagonal, taking them in the natural order ; thus : This product is called the principal term of the determinant. Now permute the subscripts in this principal term in every possible way, leaving the superscripts undisturbed. To such of the n ! resulting terms as involve the even permutations of the subscripts give the positive sign ; to those involving the odd * This notation was first employed by Cauchy in 1815. See Dostpr's Th6orie des determinants, Paris, 1877. 12 DETERMINANTS. permutations, the negative sign. The algebraic sum of all the terms thus formed is the determinant represented by the given array. Art. 7. Examples of Determinants. Applying the process above explained to the array of four elements gives a; a^' ~a;a^' - a^a^'. (i> As an example of a determinant of nine elements, with its ex- pansion, may be written < a:' a:" - a;a;'a;" - a: a:' a;" - a^a^'a,'". (2) It is evident, from the mode of its formation, that each term' of the expansion of a determinant contains one, and only one, element from each column and each row of the array. It follows that every complete determinant is a homoge- neous function of its elements. The degree of this function, with respect to its elements, is called the order of the deter- minant. Thus, (i) and (2) are of the second and third order respectively. The definition of a determinant given in the preceding article is once more illustrated by the following example of a determinant of the fourth order with its complete development : ^1 ^1 ^1 ^1 a, b, c^ d. ^ + <^^b^c^d^ — afi^c^d^ — ajb^c^d^ + afi^c^d^ ^ + afi^c^d^ — ap^c^d^ — aj)^c^d^ + ajb^c^d^ -\- ajb^c^d^ — aj?^c^d^ — ajb^c^d^ -\- aj).^c^d^ -\- aj)^c^d^ — ajb^c^d^ — aj)^c^d^ + ^■pK'^^d^ -\- ajb^c^d^ — ajb^c^d^ — ajj^c^d^ -\- afi^c^d^ (3) NOTATIONS. la It will be noticed that, in this case, the columns are ranked alphabetically instead of by the numerical values of a series of indices. Art. 8. Notations. Besides the notations already employed, the following is. very extensively used : . . a^ This is called the double-subscript notation ; the first subscript indicating the rank of the row, the second that of the column. Thus the element ^3:53 is in the second row and the third column. The letters are sometimes omitted, the elements being thus represented by the double subscripts alone.* Instead of writing out the array in full, it is customary, when the elements are merely symbolic, to write only the prin- cipal term and enclose it between vertical bars. This is called the umbral notation. Thus, the determinant of the «th order is written («) or, using double subscripts. > ^UH I These last two forms are sometimes still further abridged to («) and '^pectively. Prob. 5. Write out the developments of the following determi- 11 ants: (i) a,b, ; {2) P'P" ; (3) p> g' ; (4) a b "^K /?" P" g" a/? * Leibnitz indicated the elements of a determinant in this same manner,, though he made no use of the array. li DETERMINANTS. (S) 1 '^A^, 1 ; (6) P'P"P"' r'/' r'" ; (7) /' q' r' p" q" r" p"'q"'r"' ■; (8) {9) I II» 22 a b c a Py X y z Jo) I ^1,3 h (11) I ^o'^^«. I ; (12) I a,,a,,a,,a,. Prob. 6. How many terms are there in the development of the determinant | a^^ \ ? In the above determinant tell the signs of the terms : (3) a:a:'ar'xr«'. Prob. 7. Show that in the expansion of any determinant, all of whose elements are positive, one half the terms are positive and one half negative. Prob. 8. In determinants of what orders is the term containing the elements on the secondary diagonal (called the secondary term) positive ? Prob. 9. What is the order of the determinant whose secondary term contains 10 inversions ? 36 inversions ? Prob. 10. In, the expansion of a determinant of the «th order, how many terms contain the leading element ? Art. 9. Second and Third Orders. Simple rules will now be given for writing out the expan- sions of determinants of the second and third orders directly from the arrays by which they are represented. To expand a determinant of the second order, write the product of the elements on the principal diagonal minus the product of those on the secondary diagonal, thus : = ad — be. = -3 + 1 The following method is applicable to determinants of the third order :* * This method was first given by Sarrus, and is often called the rule of Sarrus; sec Finck's 6l6ments d'Algebre, 1846, p. 95. Likewise, a b c d -9 5 2 il = - 3 + 10 = 7. SECOND AND THIRD ORDERS. 15 Beneath the square array let the first two rows be repeated in order, as shown in the figure. Now write down six terms, each the product of the three ele- ments lying along one of the six oblique lines parallel to the diagonals of the original square. Give to those terms whose ele- ments lie on lines parallel to __. the principal diagonal the posi- tive sign ; to the others, the - negative sign. The result is the required expansion. Ap- plying the method to the determinant just written gives After a little practice the repetition of the first two rows will be dispensed with. The above methods are especially useful in expanding determinants whose elements are not marked with indices, or in evaluating those having numerical elements. No such sim- ple methods can be given for developing determinants of higher orders, but it will be shown later that these can always be resolved into determinants of the third or second order. Prob. II. Develop the following determinants: (i) (4) (7) a k g ; h b f g f c X, y, I I cos OL cos Oi I (^) (5) (8) —n —m » (3) A c b > n — / c Ba m I b a C I P Q ; (6) cos a sin /? cos a sin /? sin a cos ^ sin a cos ^ (9) 1 v_ I > a b c cab b c a • 4 V-2 (l) Prob. 12. Evaluate the following: 1 2 3 3 I 2 2 3 I (2) - 2 o 12 2 \ 2 o 2 I (3) — I 4/"^=~i -|/' V- (Ans, — I ;; 16; 2.) 16 DETERMINANTS, Art. 10. Interchange of Rows and Columns. Any term in the development of the determinant | ^/"^j may- be written ±a^ a!' a!" .,.a^, in which ^2/'. . ./is some permutation of the subscripts i, 2, 3,. . .«. Designate by u the number of inversions in hij . . . /. Also, let V be the number of interchanges of two elements necessary to bring the given term into the form ± «/^) a^'f^ a,^"-^ . . . a^^'\ in which the subscripts are arranged in the natural order, while pqr . . . / is a certain permutation of the superscripts ', ", '" , . . . ^^'>, This permutation is even or odd according as v is even or odd. But u and v are obviously of the same class ; that is, both are even or both odd. Hence the permutations hij . . . / and pqr . . , / are of the same class ; and the term will have the same sign, whether the sign be determined by the class of the permutation of. the subscripts when the superscripts stand in the natural order, or by the class of the permutation of the superscripts when the order of the subscripts is natural. It follows that the same development of the determinant array will be obtained if, instead of proceeding as indicated in Art. 6, the superscripts of the principal term be permuted, the subscripts being left in the natural order, and the sign of each of the resulting terms written in accordance with the class of the permutations of its superscripts. Passing from one of these methods of development to the other amounts to the same thing as changing each column of the array into a row of the same rank, and vice versa. Hence, a determinant is not altered by changing the columns into cor- responding rows and the rows into corresponding columns. Thus : a, a^ . . . a^ a,' a,'' . . . ^Z'') a, a. (n) a' a" ^ (") a. («) n i*') n C') ar' a. TWO IDENTICAL PARALLEL LINES. IT Whatever theorem, therefore, is demonstrated with reference to the rows of a determinant is also true with reference to the columns. The rows and columns of a determinant array are alike called lines. Art. 11. Interchange of Two Parallel Lines. If any two parallel lines of a determinant be interchanged, the determinant will be changed only in sign. For, interchanging any two parallel lines of a determinant array amounts to the same thing as interchanging, in every term of the expansion, the indices which correspond to these lines. Since this changes the class of each permutation of the indices in question from odd to even or from even to odd, it changes the sign of each term of the expansion, and therefore that of the whole determinant. It follows from the above that if any line of a determinant be passed over in parallel lines to a new position in the array the new determinant will be equal to the original one multi- plied by (— i)"'. The element a^^'^ may be brought to the leading position by passing the ^th row over the {k — \) preceding rows, and the ^th column over the {s — i) preceding columns. This being done the determinant is multiplied by (- 1)*- . (- ly- = (- ir% which changes its sign or not according as {k-\-s) is odd or even. The position occupied by a^''' is called a positive position when {k + s) is even ; a negative position when {k -j- s) is odd. Art. 12. Two Identical Parallel Lines. A determinant in which any two parallel lines are identical is equal to zero. For the interchange of these two parallel lines, while it 18 DETERMINANTS. changes the sign of the determinant, will in no way alter its value. The value then, if finite, can only be zero. Art. 13. Multiplying by a Factor. Multiplying each element of a line of a determinant by a given factor multiplies the determinant by that factor. Since each term of the development contains one and only one element from the line in question (Art. 7), then multiply- ing each element of this line by the given factor multiplies each term of the development, and therefore the whole deter- minant, by the same factor. It follows that, if the elements of any line of a determinant contain a common factor, this factor maybe canceled and written outside the array as a factor of the whole determinant ; thus : in a. = m A determinant in which the elements of any line have a common ratio to the corresponding elements of any parallel line is equal to zero. For this common ratio may be written outside the array, which will then have two identical lines. Its value is therefore zero (Art. 12). A determinant having a line of zeros is equal to zero. Art. 14. A Line of Polynomial Elements. A determinant having a line of elements each of which is the sum of two or more quantities can be expressed as the sum of two or more determinants. Let a. (^,-^/ + ^/'±...) ^. ^3 (^3-^/ + ^3"±...) ^3 (I) be such a determinant. Then, if COMPOSITION OF PARALLEL LINES. 19" any term of the expansion of the determinant A is ± Uh BiCj . . . = ±a^ diCy. , . ^ U), bl Cj . . . ± aj, bl' ^y . . . ± . . . (2) The terms in the expansion of A are obtained by permuting the subscripts h, i,j\ ... of a^ Bi Cj . , , , But permuting at the same time the subscripts of the terms in the second mem- ber of (2), and giving to each term thus obtained its proper sign, there results which proves the theorem. Art. 15. Composition of Parallel Lines. If each element of a line of a determinant be multiplied by a given factor and the product added to the corresponding ele- ment of any parallel line, the value of the determinant will not be changed ; thus: '1« This will appear upon resolving the second member into two determinants (Art. 14), one of which will be the given de- terminant, while the other, upon removal of the given factor, will vanish because of having two identical lines. In like manner any number of parallel lines may be com- bined without changing the value of the determinant, care being taken not to modify in any way the elements to which are added multiples of corresponding elements from other parallel lines. For example, | ^,^„ | is equivalent to 20 DETERMINANTS. Art. 16. Binomial Factors. A determinant which is a rational integral function of a and of by such that if b is substituted for a the determinant vanishes, contains {a — ^) as a factor. For example, A^ a^ — p"^ a — q a-\-r b'-p' b-q b + r p q r as divisible by {a — b). To prove this, let the expansion of any such determinant be written in the form the coefficients m^, m^, m,, . . . being independent of a. Now when b is substituted for a the determinant vanishes. Hence, o = ;//o + m^b -\- mj}^ + • • • Subtracting this from the preceding gives A = m^{a — ^) + m^a^ — 3') + . . . This being divisible by {a — b), the theorem is proven. Prob. 13. Prove the following without expansion : (i) = o; (3) (4) O —X my o — mnz nz b -\- c a a b c -{- a b c c a -\-b b' -^c" (2) = 2 a b c «' + ^' o c b = 2 o — c b —a c b o a a o c - b o a = o; (5) a s'mA b sinB c sin C b - c c — a a — b o, the elements referring to the triangle ABC. CO-FACTORS : MINORS. 21 Prob. 14. Prove that I X —a y IX -a y, IX,- a y. b := 1 X y = b I Xj y^ b ix,y. IX y Qx^ — X y^—y ox-xy,^y Prob. 15. Find the value of d in the equation = o. (Ans. d = 7r/4.) sin 6^ sin 6^ o .101 o cos 6 cos Prob. 16. Show that the proportion a\b\'.l\m may be written — o; and from the properties of this determinant in the form Ifn prove the common theorems in proportion. Prob. 17. Show that the determinant ab c' c' a^ be a^ b' b' ca contains the factor {be -\- ea ■\- ab). Prob. 18. Resolve the following determinants into factors:* (i) (4) Art. 17. Co-factors; Minors. The terms oi A = \ «/"' | which contain the element a^ may I a a^ lb b' > (2) la a" lb b' a' b^ 1 (3) 1 a^ a la^a ; . . . ^, n-i U-I le c" le c" e' id d'd' lUnan . , . an""-' I I a b a^b'' a* b' I I c d c' d' c' d* > (5) I a a' I I b e b'e' > (6) III a b e a' b' c' be obtained by expanding the determinant a, a, a. ,a. («) a' a" a'" a («> (1) For, in writing out this expansion each term is formed by taking one, and only one, element from each column and each * These determinants belong to an important class known as alternants. See Hanus' Elements of Determinants, Boston, 1888, pp. 187-201. 22 DETERMINANTS. row of the array (Art. 7). If, therefore, in selecting the ele- ments for any term, any other element than a/ be taken from the first column, the one taken from the first row must be zero. Hence, the only terms which do not vanish are those which contain the element a/. Moreover, in the terms of the expansion of (i) which do not vanish, a/ is multiplied by {n — i) elements chosen one from each column and each row of a'' a. .a„ an . a, (n) (2) There are {n — i)! such terms, any one of which may be written ± a^al'aj" . . . ^Z**^ ; the sign being determined by the class of the permutation of the n subscripts i, i,j, . . . I. But since this is of the same class as the permutation of the {n — i) subscripts /,/, . . . /, the sign of any term, ± a^aCaj" . . . ai^''\ of the expansion of (i) is the same as the sign of the corre- sponding term, al'a-" .. .a^''\ of the expansion of (2). H^nce, <2/ o o .. .0 .a. -(«) {n) («) (3) The determinant (2) is called the co-factor or complement of the element «/ in the determinant |<^/"^|. It is obtained from this determinant by deleting the first column and the first row. The co-factor of any element a^^^'^ maybe found in the same manner upon transposing this element to the leading position. But by this transposition the sign of the determinant will be changed or not according as a^''^ occupies a negative or a posi- tive position (Art. 11). Hence, to find the co-factor of any element ^/*^ of the determinant l^/**^], delete the row and the column to which the element belongs, giving the resulting determinant the ) V^^ | sign when (^ + .) is { -" } . DEVELOPMENT IN TERMS OF CO-FACTORS. 25 The co-factor thus obtained is represented by the symbol the sign-factor of which, (— i)*+^ is intrinsic, i.e., included in the symbol itself, which is accordingly written as positive. The co-factors of the various elements of | a^^a^^a^^ \ are as follows : ^n = ^38 ^33 i ^n-- ^ai ^a3 ^31 ^33 y A^^ = ^,1 ^aa **^3i **3a A.. = - J ^aa = ^n ^13 ^3. ^33 » -^83 — ^n ^:a ^31 ^3a A,,= f ^3a^- ; -^33 ~ ^ax ^aa The result obtained by deleting the k\ki row and the jth column of J ^ I ^/"^ | is called the minor of the determinant with respect to the element dk't and is written A%. This minor is the same as the co-factor of the same element without its sign-factor ; thus : Similarly A\i\k is the result obtained by deleting the ^th and ^th rows and the /th and ^th columns of z/, and is called a second minor of the given determinant. Minors of still lower orders are obtained in a similar manner, and expressed by a similar notation. The >^th minors are determinants of the order {n — k). Art. 18. Development in Terms of Co-factors. The {n — i)\ terms of | <3!/"^ | which contain a^'^ are repre- sented in the aggregate by a^^'^Ak^'^ (Eq. 3, Art. 17). In like manner the groups of terms containing the successive elements ^k, ^k' y ' • • ^A^"^ are respectively Each one of these n groups includes (n — i) ! terms of the determinant | ^/"^ | , no one of which is found in any other 24 DETERMINANTS. group. In all of them, then, there are nX{^ — i)l or n\ dif- ferent terms of the determinant, which is the whole number. Hence, («) I = aM/ + «."^/' + . . . + a.^'^W'. Similarly (Art. lo). («) «/^M/^> + a.^'^A,^'^ + . . . + a„^'^A„^^\ (I) (2) Any determinant may, by means of either (i) or (2), be re- solved into determinants of an order one lower. Since, in these formulas A^y . . . Ak^"l or A^^'\ , . . A^'^ are themselves determinants, they may be resolved into determinants of an order still one lower in the same manner. By continuing the process any determinant may ultimately be expressed in terms of determinants of the third or second order, which may be easily expanded by methods already given (Art. 9). For example, let it be required to develop the determinant ^ = \ a^ d^ c, d^ \ , Applying formula (i), letting k= i, gives A = a^ Upon a second application of the same formula this becomes b, c. < - b. a, c, d. + ^. a^ b, d. -< a^ b, c^ b^ C, d. ^3 ^9 < a, b, d. ^s b, c. b, C, d. a, c, d. ct, b^ d. a, b^ c. = a,b. c,d. - a,c. Kd, + a,d. b.c. c,d. b,d. b,c. -a,K c,d. + ^^, a,d. -b,d. a, c. c,d. a,d. a, c. + a,c. — a,d. I'.d, -Kc, a,d. + c, d. a,b. b.d. a J, a, b. b,c. + b,d, a,c. -c,d. a>b. b.c. a,c. aj. The complete development may be written out directly from the above. It is given in Eq. 3, Art. 7. THE ZERO FORMULAS. 25 Prob. 19. Develop the following determinants: (i) I jc 17 X I y I ly 1 X y I X I (2) a X y a X 6 o y y Q o X a y X a (3) o q r s p o r 5 p q o s pqro Prob. 20. Find the values of the following determinants: (i (4) 1234 2341 3412 4123 I I I 1 o I I I I O I I I I o (2) (s) 0102 1020 0201 2010 (3) 3 6 9 -3 8 3 5 3 6 -I 5 3 3 3 3 3222 2 2 I I I I I o (Ans. 160; 9; o; (6) 0003 1002 O I O I 0010 3; -3; -3) Prob. 21. Obtain the determinants in Exs. 5 and 6 of the pre- ceding problem from that in Ex. 4. Prob. 22. Evaluate oil. I o I . I I o. , of the «th order. (Ans. (« - i)(- i)«-' ) Prob. 23. Show that bed a —d c d a —b — d —c = {a'-^-b'+c'+d'y. Art. 19. The Zero Formulas. If in the determinant | ^/"^ | the ^th and y^th rows be sup- posed identical, the elements ^/, ^/', . . . a^'^'^ in the formula (i) of the last article may be replaced by «/, ^/', . . . U},^'''^ re- spectively. But in this case the .value of the determinant is zero (Art. 12). Hence, in reference to the determinant I a^""^ I , h and k being different subscripts, aj!A^ + a,''A^' + . . • + a.^^^A,^-^ = o. 26 DETERMINANTS. Similarly, p and s being different superscripts, ^^w^^w + a^f^Ai^^ + . . . + ai^^A^^^^ = o. Art. 20. Cauchy's Method of Development. It is frequently desirable to expand a determinant witli reference to the elements of a given row and column. Let the determinant be ^ eee | a^^"^ \ , and the given row and column the ^th and /)th respectively. Then is A^^^^ the co-factor of «/^\ the element at the intersection of the two given lines. The co-factor of any element aj^^'^ of A^^^ will be designated by Bf,^'\ this being a determinant of the order (n — 2). The required expansion may now be obtained by means of the following formula, due to Cauchy : I ^/«) I = a.^'^Ai^^- ^at^ai^^B,^^ (i) in which k = i, 2,. . .^ — i, ^ -f- I, . . . «, and J = I, 2, .. ./ — i, /+ i» ••• ^> successively. To prove this, consider that B^'^ is the aggregate of all terms of the expansion of A which contain the product ai^^ai'\ These terms are included in a,i^^Aj}^\ Now, every term in the expansion which does not contain a^^'^ must contain some other element a,i''^ from the ^th row and also some other element a^^'^ from the /th column, and thus contains the prod- uct alf^ai*^. But this product differs from ai^'^Uj,^'^ only in the order of the superscripts ; and is, therefore, in the expansion of A, multiplied by an aggregate of terms differing in sign only from that multiplying ^^^^W^. Hence, — a^^^W^Bk^''^ is the coefficient of a^'^'a^^^ in the required expansion. In the formula a^^^Ai^^^ gives (n — i) ! terms of J. There are also (« — i)" such aggregates as — aJ^^^a^^'^Blf^^ each con- taining (« — 2) ! terms. The formula therefore gives {n — \y,-\-{n — \f {n — 2)\ = n \ terms, which is the complete expansion. When the expansion is required with reference to the ele- CAUCHY S METHOD OF DEVELOPMENT. merits of the first column and the first row the formula, written explicitly, becomes {^ W| = a; a:- a;a;'B;' - a:al"B;" - ... - a^a^-^B^-^ - a^a^'B;' - a:a;"B^" - ... - a^a^^^B^-^ a:a^^^B^^\ (2) in which B^^'^ has intrinsically the sign (— \f^\ Cauchy's formula is particularly useful in expanding deter- minants which have been bordered ; such as -Q = u. 7^2 «, «1 «,, ^,,^,3 u. ^.I ^22 ^« u. ^3. ^32 ^33 (3) Applying formula (2) to this determinant gives - fi = - «■' ^23 ^,3 + «,«, a.. ^« - 7/3^^, ^21 ^35 ^32 ^33 ^3. ^33 ^3, ^3, ^1, ^,S - ^.' ^n ^13 + M,U, ^n ^1, ^8, ^33 ^3, ^33 ^31 ^35 ^1, ^,3 ^u,u. ^n ^,3 -K ^n ^n ^« ^.3 ^21 ^23 ^31 ^,Q Letting ^^^ = ^,^, and, writing ^4,1, ^j,, ... for the co-factors of the elements of | ^1,^5,^3, | , the above becomes Prob. 24. Develop the following determinants by Cauchy's formula: (i)- a h g u hb fv gfcw u vw o (2) yz zx xy ; (3) yz I I zx I 1 xy 1 1 I I I 1 o xyzx I .Tjv oyz I zx yz o 28 DETERMINANTS. (4) — I — X I I ; (5) 1 1 I X ; (6) O a b I - y — I I X y z o — a sin^sin^ X o y z 1 1 ly -b- - cos^ cos^ I — z I — I I I I z Art. 21. Differentiation of Determinants. By the formula (i) of Art. i8 A ~ I y,^^ I = Yk.yu, + Yk^yk^ + . . . F^^/^. (i) Considering the elements of the determinant as independent variables and differentiating with respect to yk^ gives Substituting in (i), Similarly d/i or dA Y -^ =^-^^^^+-^^'^+ +J. dA dy, kn (2> (3> SA (4> Again differentiating (i), this time with respect to all the ele- ments of the k\.\\ row, there results S,A = Y,4y,. + Y,4y,, + . . . + YkJy,^- (5> In the total differential of A there are obviously n such ex- pressions as (5), each of which may be obtained from A by replacing the elements of some one of the rows by their differ- entials ; thus : dA dy,, . . . dy^„ f.r "• y^n fm ' ' ' ynn + -y.n ,dy. Jn, y. + ...+ 7n y.^ dy,,^ .,.dy,^ .(6> If all the elements are functions of one independent variable Xy, then, representing —^ by j/^/, dx dA dx 7i, y^x y.n y.n ynn + 7a/-. • yJ yr yr +...+ J.I y^n y.n -ynn .(7) RAISING THE ORDER. 29 Prob. 25. Show that Cauchy's formula may be written dau'^'^ dak'^^da^- Art. 22. Raising the Order. Since, in the expansion of the determinant (i) of Art. 17 the elements a^, , . . a^ do not appear, these may be replaced by any quantities whatever, as Q, . , , Ty without changing the value of the determinant ; thus: ^, o o . . . o a: a:' a:" . . . ^,(«) ^/ o o . . . o Similarly, ^/ a J aJ' . .. ... =«' . . «,"" al ... Qa^' ... M'J l-^J \J a:". . . «i"' a: a:' a:" . . . «„'"' TNaJ"...ai''^ in which Q, R, , . . T and Z, . . . iVare any quantities whatever. Finally, o =«'... ^«ir^W"^= < o ... o o o QaJ'... o o a^ o . . . o a^ a^ . . . o ^'«-x^Vx...^jr^o ^« ttn . . ^ <«-^> ^ <«> s M . . .a^^:^r^ o that is, if all the elements on one side of the principal diagonal are zeros'the determinant is equal to its principal term, and the elements on the other side of this diagonal may be replaced by any quantities whatever. By what precedes, / . . . ^ .<«> «„'... a («) 10... Qa:... «,<"> Taj... «»"" 30 DETERMINANTS. Hence, a determinant of the nth. order may be expressed as a determinant of the order (« -|- i) by bordering it above by a row (to the left by a column) of zeros, to the left by a column (above by a row) of elements chosen arbitrarily, and writing i at the intersection of the lines thus added. By con- tinuing this process any determinant may be expressed as a determinant of any higher order, Prob. 26. If all the elements on one side of the secondary diag- onal are zeros, what is the value of the determinant ? Prob. 27. Develop the determinant a h g u h b f V S f c w u V w t / s Prob. 28. A determinant in which ^^^^^ = — as^^^ and ak [k) O IS said to be skew-symmetric. Prove that every skew-symmetric deter- minant of odd order is equal to zero. Art 23. Lowering the Order. The following method of reducing and evaluating a deter- minant is often useful, particularly when the elements are numer- ical. Let the determinant be J = ai'a/' ai in) J = I Then («iO /\n-l ^2 0L\ ^2 Ci\ (I2 «3' ai'as" ai'aa"' («i') An-2 (1\ OL2 — ^2 ^1 ^1 ^2 — ^2 t*! a^ar (Art. 13.) (Arts. 15, 17.) In this last expression the determinant is obviously of the order (w — i). The process may be formulated thus: Replace each element a)^^^ of the minor of the leading element by the SOLUTION OF LINEAR EQUATIONS. 31 determinant and the resulting determinant divided by (ai')'»~2 will be equal to the given determinant. The elements being numerical the process may be repeated with ease until the order becomes unity. The example given below will illusffat 1234 = I 2 I 3 I 4 8765 87 8 6 85 1827 3645 I 2 I 8 I 3 I 2 I 4 I 7 I 2 I 3 I 4 36 3 4 3 5 -9 -18 -2) ' = -9 -18 - -9 -27 6-1 . ' 6 - I 6 3 - 5 - / -9 -18 - -9 -27 1 - 5 - 7 117 135 = 1296. 45 63 Art. 24. Solution of Linear Equations. Of the many analytical processes giving rise to determinants the simplest and most common is the solution of systems of simultaneous linear equations. Thus, solving the equations a.V + a:'x" = /<•„ ) a:x' + a,"x" = K,, \ by the methods of ordinary algebra gives : X' = In the notation of determinants these are written : K,a,'' - K,a/' x" «i «t — c'i ^t « X = / a, a- / It will be noted that the two fractions expressing the values of x' and x'^ have a common denominator, this being the de- terminant whose elements are the coefficients of the unknowns arranged in the same order as in the given equations- The 32 DETERMINANTS. numerator of the fraction giving the value of x' is formed from this denominator by replacing each coefificient of x' by the corresponding absolute term. Similarly for x" . The difficulty of solving systems of linear equations by the ordinary processes of elimination increases rapidly as the num- ber of equations is increased. The law of formation of the roots explained above is. however, capable of generalization, being equally applicable to all complete linear systems, as will now be shown. Let such a system be written a;x' + a^'x" + . . 4- <3:/«';ir(«' = K^ , a:x' + a^'x" + . . . + ar'x^''' = k„. (I) Now form the determinant of the coefficients of these equations ; thus : n ,a, («) and let A^^ be the co-factor of ai''^ in this determinant. The function is equal to D when p — s (Art. i8) ; to zero when / and s are different superscripts (Art. 19). Then, multiplying the given equations by y4/*\ A^""^^ . . . Aif^ respectively, the sum of the resulting equations is a linear equation in which the coefficient of x^^^ is equal to D, while those of all the other unknowns vanish. The sum is, therefore. Dx^'' = /c,^/^> + K,Ai^^ + . . . + K^A, (2) But the second member of this equation is what D becomes upon replacing the coefficients «/*\ a^^^, . . . a^f^ of the unknown ;ir<*' by the absolute terms /c, , /c, , . . . /c« in order. Hence, -W — a/. CONSISTENCE OF LINE / SYSTEMS, a.'a," . . 33 (3) a.'.. .aJi^-^^K^ai^^^K,.a^^^ «>.".. • «„'"' This result may be stated as follows : {a) The common denominator of the fractions expressing the values of the unknowns in a system of n linear equations involving ;/ unknown quantities is the determinant of the coefficients, these being written in the same order as in the given equations, {b) The numerator of the fraction giving the value of any one of the unknowns is a determinant, which may be formed from the determinant of the coefficients by sub- stituting for the column made up of the coefficients of the unknown in question a column whose elements are the absolute terms of the equations taken in the same order as the coeffi- cients which they displace. Prob. 29. Solve the following systems of equations : (l) 3^ + 5^=21. 6^ + 27=15; (2) f + f = s, f+. = .= (3) 3^ 4- 7 + 22 = 50, ^ + 2j; - 32: = 15, (4) I 1 I ^ ^ I ^ ^ I ^ (5) w , X . y . z ^ w . X w , X . y . z 7 + 9 + " + ^ = ''^^' -+^ = 2144, - + - + ^ + - = 1472. 9 II 13 15 Prob. 30. Show that the three right lines y = x-\- I, y = — 2x + i6y y = sx — gy intersect in a common point. Art, 25. Consistence of Linear Systems. When the number of given equations is greater than the number of unknowns their consistency with one another must 34 DETERMINANTS. obviously depend upon some relation among the coefficients. This relation will now be investigated for the case of {n -\- i) linear equations involving n unknowns. Let the equations be ^,V + ... + a. W^yW _ «■.. ^ ««v + ... + a. («) -yin) ^n\ ^«+x V + . . . + ;ir(«> = O. Representing the determinant of the coefficients by E, the general solution, as given by the formula (3) of Art. 24, is ;ir(^) = O/E, That is, all the unknowns are equal to zero, and the equa- tions have no other solution than this unless E = o. 36 DETERMINANTS. But in this case the value of each unknown is obtainable only in the indeterminate form o/o. The ratios of the un- knowns may be readily obtained, however. For, dividing each equation through by any one of these, as x^'\ the system (i) becomes (*-i) I- (2) a'- I- _i-^(*-i)f _L^(s+i)::r l 4_/7(«)_ — _/7(^) • ^>+ ■■■+-• is -I) X^^ +a is+iY '(S+l) is) + ...+a (n) ;r<'') ,(s) J Now the condition E = o establishes the consistency of the n equations (2) involving the {n — i) unknown ratios (Art. 25), -(*)' .(s+i) An) Is)' Hence, if ^ = o'the given equations (l) are consistent ; that is, the values of the above {n — i) ratios obtained by solving any (n — i) of them will satisfy the remaining equation. Any n -quantities having among themselves the ratios thus determined will satisfy the given equations. Thus, if x^\ x^\ . . . x^'^^ are n such quantities, so also are \x^, A x^' , . . . Ajf/'*^ A being any factor whatever. The determinant E of the coefificients of the given homo- geneous linear equations is called the resultant or eliminant of the system. When the number of equations is greater than the number of unknowns the conditions of consistency are expressible in the form of a rectangular array, as in Art. 26. As an example, consider the five equations 2x- iy^z-=^o, 4x -j/ — ^ = o, - 7-^ + 37 + ^ = o, x-]-}> — js = o, t^x — ^y -\- z = o. Dividing each of the first two equations by z and solving X y Tor the two unknowns - and - gives z z — I I 2 — I 14 I 2-3 4— I ^3^ 5 CO-FACTORS IN A ZERO DETERMINANT. 37 or x:j/:2::2:siS'i and any three quantities having these ratios will satisfy the two equations, as lo, 15, and 25. That the third equation is consistent with the first two is shown by the vanishing of the determinant 2 — 3 I 4- I - I -73 I If all the equations are consistent the determinant of the coefficients of any three of them must vanish ; that is, 2 4-7 I 5 -3 - I 3 I -5 I — I I — I I Art. 28. Co-factors in a Zero Determinant. If, in the preceding article, £ = o, it follows from Arts. i8 and 19 that a/ A,' + a," A," + ...+ are proportional to ^/, A^' ^ . . . Af'^, whatever the value of k. Thus, giving to k the successive values i, 2, . . . n, there result x' \x" \.,, :x^"^::A/:A,'':. . . i^/'*) i'.AJ'.A:':... :^,<«\ : : AJ : A, AJ-l 38 DETERMINANTS. Hence, when a determinant is equal to zero, the co-factors of the elements of any line are proportional to the co-factors of the corresponding elements of any parallel line. Art. 29. Sylvester's Method of Elimination.* Let it be required to eliminate the unknown from the two equations a^x^ + ^i^^ -\- a^x -{- a^ = o, d,x' + b^x -\-b,-=Q. This will be done by what is called the dialytic method, the invention of which is due to Sylvester. Multiplying the first of the given equations by x^ and the second by x and x"^ suc- cessively, the result is a system of five equations, viz.: a^x^ + a^x"" -)- a^x -}- ^„ = o, b^x'' + b^x' + b,x = o, b,x\+ b,x' + b,x' =0. The eliminant of these five equations, involving the four unknowns x, x\ x\ and x* is (Art. 25) £ = a. ^. a^ ^0 a. ^h «, ^0 K K ^ K b^ K K b. K = 0. If the given equations be not consistent this determinant will not vanish. The above method is a general one. Thus, let the two given equations be a^x^^ + . . . + /3!,;ir + ^„ = o, ^„^"-f + ^,^ + ^0=0. Multiplying the first equation (;2 — i^ times in succession by x^ and the second {m — i) times, {m -\- ti) equations are * Philosophical Magazine, 1840, and Crelle's Journal, Vol. XXI. SYLVESTER S METHOD OF ELIMINATION. 3^ obtained which involve as unknowns the first {m -\- n — i) powers of x. The eliminant of these equations is a determinant of the order {m -\- n), which is of the nth degree in terms of the coef^cients of the equation of the mth degree, and vice versa. The law of formation of the eHminant is obvious. The same method may be used in eliminating one or both the variables from a pair of homogeneous equations. As an example, let it be required to eliminate the variables from the equations 2x^ — ^x^y — 9jj/^ = o and '^x'^ — yxy — 6y = o. X Dividing the first byjJ/^ and multiplying by — ; the second X byy, and multiplying by— twice in succession, there result, X X X X in all, five equations involving — , — , -^, and -j- these four ratios gives Eliminating E~ 5 o- 9 — 9 o 3-7-6 7 — 6 o 600 the vanishing of which shows that the two given equations are consistent. Prob. 31. Test the consistency of each of the following systems of equations: (i) ^+^+22=9, x-\-y — z=o, 2x—y + z=^, x—3y+2z=i; (2) X —y — 2Z =0, X — 2y •\- z = o, 2X sy o; (3) 2^y — ^y = o, 8^ jV + Sxy' — 5/ = o. Prob. 32. Find the ratios of the unknowns in the equations 2X -{-y — 2z = o, 4W — y ^ 4Z = o, 2w -{- x — ^v -\- z = o. Prob. 33. In the equations ak'x' + . . . +«a('')jc(«) + ^^(« +^)^(«+i) r= o, [^ = I, 2, . . . «] prove that ^' : . . . : x^""^ : ^(«+^> :: J/' : . . . : J/^«> : J/(«+i) , where 4-0 DETERMINANTS. {— i)'"'-^^'^ is the determinant obtained by deleting the zth column irom the rectangular array M = aWa^(n+^) an . . , Un ^H lx-\-yy -\- i^z _vx -\- my -\-\z _ ^x-\-\y + nz P ~ ■ X y _ y /// I ^p Iv p m\ q V \ q V m q \ n r }J. n r )xX r Prob. 34. From deduce Prob. 35. Show that the three straight lines dt':r -|- b'y -f- / = o, ta^'x + b"y + /' = o, and a"'x + b'"y + c'" = o, are concurrent when I ^rV" | = o. Prob. 36. Prove that the medians of a triangle are concurrent. Prob. 37. Show that the points (x^,y^), (x, ,}>,), and (x^,y^) are collinear when x^ y^ i = o. •^0 y. I X, y^ I ^, y. I Prob. 38. Write the conditions that all the points (^,,jKi), ■{x^,y^), . . . (xn,yn) shall be collinear in the form of a matrix. Prob. 39. Obtain the equation of a right line through (x^^yj .and (x^,y^) in the form of a determinant. Prob. 40. Show that the equation x y z j X, y, z^ I x^ y, z^ I ^3 y, ^3 I 'represents a plane through {x^ , y^ , z^), (x^ , y^ , z^), and (^g , y, , z^). Art. 30. The Multiplication Theorem. Let the two homogeneous linear equations .•be subjected to linear transformation by substituting (I) (2) THE MULTTPLICATION THEOREM. 41 (3) (4) The result of such transformation is (a J,, + aj.^u, + {aj^, + aj^^u. The vanishing of the determinant is the condition that the equations (3) may be consistent ; that is, the condition that they may have solutions other than u^ =z o =z u^ (Art. 27). Now the equations (3) may be consist- ent because of the consistency of the equations (i), in which case the determinant «.. a,, , (5) vanishes. Or, this condition failing, and the equations (i) thus having no solution other than x^=z o ^= x^y the equations (3) will still be consistent if the equations (2) are so ; that is, if the determinant ^11 ^X1 (6) vanishes. The vanishing of either of the determinants (5) or X6), therefore, causes the determinant (4) to vanish. It follows that (5) and (6) are factors of (4) ; and since their product and the determinant (4) are of the same degree with respect to the coefficients When the two determinants to be multiplied together are of different orders the one of lower order should be expressed as. a determinant of the same order as the other (Art. 22), after which the above rule is applicabje. The product of two determinants may be formed by columns, instead of by rows as above. In this case the result is obtained in a different form. Thus the product of the de- terminants (5) and (6) by columns is Prob. 41. Form the following products : (3) (i) la A g h b f g f c ^1, ^12 ^.3 ^21 ^22 ^23 ^»1 ^32 '^SS \g c (2) b f f c ^ g g c a h h b ^)2 ^13 A 4 4 4 21 22 23 -^31 -^32 -^33 ; (4) a, b, c. . oil a, b, c^ I I «3 K ^3 I I Prob. 42. Generalize the last example (see Prob. 22, Art. 18). Prob. 43. By forming the product a-bV- I ■\- m V — I l-\-mV- I j- k V~^^ * Carr's Synopsis of Pure Mathematics, London, 18S6, Article 570. PRODUCT OF TWO ARRAYS. 43 ishow that the product of two numbers, each the sum of four squares, is itself the sum of four squares. Art. 31. Product of Two Arrays. The process explained in the preceding article may be ap- plied to form what is conventionally termed the product of two rectangular arrays. It will appear, however, that multi- plying two such arrays together by columns leads to a result radically different from that obtained when the product is formed by rows. Let the two rectangular arrays be ^n^iQ^is and b^.b^^b^^ The product of these by columns is / ^11^12 + ^^/sQ ^i^b,^ + a^^b^^ a^J}^^ + «„^„ The determinant ^ is plainly equal to zero, being the prod- uct of two determinants formed by adding a row of zeros to one of the given rectangular arrays and a row of elements chosen arbitrarily to the other. In general, the product by columns of two rectangular arrays having m rows and n columns, m being less than Uy is a determinant of the n^^ order whose value is zero. Multiplying together the above rectangular arrays by rows, the result is A'^ ^n 6,, + a ,/.. + ^13^,3 a.Ar + a 12^22 + ' ^13 5„ ^2/11 + ^22<^I2 + ^23<^13 <3:3,/53, + ^„^„ + ^„^„ a,,a,. . 6Jn + ^,1^13 . bj.. -U ^n^i2 . ij. ^22^23 6.A, ^21^2 3 b.A. ^21^,2 b ,A, 1 In the same manner it may be shown that the product by rows of two rectangular arrays having m rows and n columns, m being less than «, is a determinant of the m^^ order, which may be expressed as the sum of the n \/m ! {n — w) [determinants 44 DETERMINANTS. formed from one of the arrays by deleting {n — m) columns,, each multiplied by the determinant formed by deleting the same columns from the other array. Art. 32. Reciprocal Determinants. The determinant formed by replacing each element of a given determinant by its co-factor is called the reciprocal o£ the given determinant.* Thus, the reciprocal of ^n^„ IS A -^ai-^23 . . . A^n The product of these two determinants is S.^= aiiAn-\- . . . -\-ainAin flii^2i+ . . .+«in^3n CliiAni'\-' . '-{-a-inAnn aniAii-j-. . . .-{-annAm «nMai+- • .+«n»'43n. • . . aniAni+. • -+annAnn Each element on the principal diagonal of this product is equal to 6 (Art. i8), while all the other elements vanish (Art, 19). Hence, d,A = 6 o. . o d .. Q(n) o = (J% or ^ o„ o . . . d That is, the reciprocal of a determinant of the n^^ order is equal to its {n — i)*^ power. Art. 33. Linear Transformations. Let it be required to transform the system ahiXi+ah2X2 + . . .+ahnOCn = o [A=I, 2, ...w] (l) into a new system with the variables Wi, W2j • • • ^n by means of the linear substitutions QUANTICS: INVARIANTS AND COVARIANTS. 45 Making the required transformation, as has been done for the case of two variables in Art. 30, the resulting system is PjlUl +pj2U2 + . . . +pjnUn = 0, [j = I, 2, ... w] (3). in which pks = c^hKi + (^k2bs2 + . . . + aknbsn- (Eq. 9, Art. 30.) The determinants of the systems (i), (2), and (3) are thus con- nected by the relation l/^l.nhl^l.nl.l^l.nl. (4) The determinant | ^i.n |> whose elements are the coefficients in the equations (2), is called the modulus of transformation; and the relation expressed by equation (4) may be stated as follows : If a system of n homogeneous linear equations in n variables be subjected to linear transformation, the eliminant of the trans- formed equations will be the eliminant of the given equations- multiplied by the modulus of transformation. A transformation whose modulus is unity is said to be unimodular. Prob. 44. Show that the following transformations are unimodular: (i) x=x'+y^+2z', y=x'-\-y'-\-z'^ z=y'-\-z'\ (2) ^=:x:' cosa— y sina, >'=x' sinaH-^'' cosa. Art. 34. Qu antics; Invariants and Covariants. A homogeneous function of any number of variables is called a quantic. A quantic is binary, ternary, . . . w-ary according as it con- tains two, three, , , .n variables; and is specifically known as a quadric, cubic, . . . w-ic according as it is of the second, third, . . . wth degree. Thus, the function q= aiiXi^ + ^22^2^ + ^33^3^ + 2a23^2^3 + 2azxXzXY + 2a\ 2^1 rv2 is a ternary quadric. A covariant is a quantic derived from another quantic in such manner that when both are transformed by the same linear sub- stitutions the resulting quantics are still connected by the same process of derivation. 46 DETERMINANTS. An invariant is a function of the coefficients of a quantic which is not effected by linear transformation of the quantic, except that it is multiplied by a power of the modulus. It is obvious that every invariant of a covariant is an invariant of the original quantic. Art. 35. The Discriminant. The discriminant of a quantic is the eliminant of its first derivatives. Thus, the binary quadric (f)= aiiXi^ + ^22^2^ + 2ai2^i:x:2=o gives 1 30 , 130 —x — = aii^i + ai2^2=o, —7^ — = ^1 2X1+^22^2=0. 2 OXi 2 0x2 Hence, writing ai2 = (i2i, the discriminant is an ^12 ^21 ^22 If 'i, >'2> • • • yn be n functions, each of the n independent variables ^i, ^2, . . . ^n- Then the determinant J^ 'byi ^y\ dX2 'dy2_ dX2 '^yn dX2 dyi 3j2 . _ oyn dXn dXn 'dXr? is called the Jacobian of the given functions. The notation J _d(yi,y2,- . -yn) d(Xi, X2,... Xn) is in common use, being suggested by the close analogy between the Jacqbian and the ordinary differential coefficient. When the functions are linear, thus: yi = auXi + a2iX2 + . . . + aniXn [i=i,2, . . .n] it follows from the above definition that the Jacobian is the -determinant of the coefficients. That is, /= ! aiia22 ... Ann I . When the functions are not independent; that is, when J^iyi) y2y ' ' ' yn)=Of the Jacobian vanishes. For, differentiating this function with respect to each of the ■variables Xi, X2, . . . Xn gives the consistent system dyi dxi 'by 2 dxi JACOBIAN OF INDIRECT FUNCTIONS. 49 and eliminating :~— , . . . ^ — from these equations there results ' dXn ^J=o. Art. 39. Jacobian of Indirect Functions. When )'i, >'2> • • • Jn are each functions of (^i, ^21 . . . (^m these being functions each of x^, ^2, . . . x^, then d(x^ X2,... Xn) ^(Cl, C2' • • • Cn) ' ^(^U ^2J • • • ^n)' This may be demonstrated by writing out each of the Jacobians in the second member in determinant form, changing columns into rows in the first, multiplying the two together by rows, and interpreting the result by means of the relation dyi_J^'^±^'^^^ , ^yi '^Cn dXk 9^1 ^^k 3C2 ^^k ' ' ' ^Cn '^^k Thus, for w = 2. d(yv y^) . ^(Cp Q d{^vQ d(x,,X2) ^ ?a , ?ii 5C2 dy, 'dy^ 9Ci 9C2 ^Ci 9C2 3G ^ ?Ci ?C2 9^2 9^2 9,^1 9^2 'd^2 ^^3 9^1 9:x;i 9(^2 ci^t^i ?C2 'dx^ d{yx, y^ d(Xi, x^y 5^2 5Ci_^?>:_2 90 9:^2 9^2 9G 9:x;- 9^1 9;yi 9^1 9^2 dxj^ dx2 It may be shown in precisely similar manner that, when the functions yi, y^, . . . yn are independent, ^(yt> yy ' " ^n) d{x^,x.,. . .Xn) _ J(rVi, :V2, . . . Xr^ ' d{y\, y.^, . . . >'„) 50 DETERMINANTS. Art. 40. The Jacobian a Covariant. When, in the preceding article, the functions ^^ C2> • • • C« ^-re linear, thus: (^i = a^iXi + a^iX^ + . . . + a^iXn ; [i = i , 2, . . . w] then (Art. 38) ^ fa> yi^'" yn) _ I ^ ^ ^^^| ^(yi>>^2>»»-y n) d\X^'i ^2> • • • -^n) ^(Ci' C2J • • • C«y Hence, if a set of functions be subjected to linear transformation, the Jacobian of the transformed functions is equal to that of the given functions multiplied by the modulus of the transformation. That is, the Jacobian is a covariant of the set of functions from which it is derived; unless these functions are linear, in which case it is an invariant. Art. 41. Jacobian of Implicit Functions. When the functions are implicit, thus: ^i{yv yv' yn, OC^, X^,,.. Xn) = 0\ [i=I, 2, . . . W] then d(yv yv'"yn) _. ^ ^ d{X,, X^, ... Xn) d(Xi, X^,... Xn) d((pij 02> ' • ' n) ' d(yv yv' yn) To prove this, write the above Jacobians in determinant form,, change columns into rows in the first member, and clear of frac- tions. This gives, representing by P the resulting product in the first member, P= dyi dxk ' ' ' 'dyn '^Xk THE HESSIAN. 51 Now, the total derivative of (j)i with respect to Xk is dXk '^Ji '^OCk ' ' ' '^Jn ^Xk But, since x^^ ^2» • • • ^n are independent, this becomes The above product thus becomes P= 9^- = (-!)" ?iXk = (-l)n which is the required proof. Art. 42. The Hessian. The Jacobian of the first differential coefficients of a function of n variables is called the Hessian of the function. Thus, the Hessian of ^(x^, x^^ . . . x^^ is ff(<^)- Since /3<^ ^ ^i>\ 8V "^Xdx: 'dx,""'dXn} ?iXn^X^ d{Xij X2, ... Xn) dxfiXn dxk'^Xs Zxgdx k the Hessian is a symmetric determinant. The Hessian of a quadric differs from the discriminant only by a numerical factor. Let the function ^ be transformed into ^' by the linear sub- stitutions Xi= a^iU^ + ^21^2 + . . . -\- ani Xn. [i=ii 2y , . .n] 52 DETERMINANTS. Then (Art. 40) But ;z — -^ — =^ — p^, and the above equation may therefore be written G^(2/p ^2, . . . Un) = a 2 CXn = \a„n\'H (), That is, if a function be subjected to linear transformation^ the Hessian of the transformed function will equal that of the given function multiplied by the square of the modulus of the transformation. It follows that the Hessian is a covariant of the function from which it is derived; unless this function is a quadric, in which case it is an invariant (Art. 35). Prob. 45. Tell whether or not the following quadrics are prime: (i) ^x^—gy^—$z^-\-i8yz+24ZX—i2xy; (2) x^+z^+yz--2zx+xy; (3) gy^+i^yz—6zx-\-8xy. Prob. 46. Find the values of X in order that each of the following quadrics may be composite: (i) Xxy-\-szx+sy^+2Z^; (2) 2X^—sh^—'^^^^+^7yz+^zx—xy; (3) /^x^+^z^-^yz+T,zx+2xy-^X{x^+Sy^—yz+szx+sxy), Prob. 47. Find the Jacobian of the functions yi = i-Xiy ^'2=^1(1-^2), >'3 = ^1^2 (1-^3), ... yn = XiX2 . . . Xn-i(l-Xn). THE HESSIAN. 53 Prob. 48. Show, 1 y means of their Jacobian, that the functions yi = {x-X2)(X2 + X2), y2={Xi-\rX2){X2-X3), Ji, = ^^ (^'c " ^1 ) are not independent. d( X "V^ Prob. 49. Find ' ; having given x=pcosd and 3'=^sin^, in which ^= aw, d = hv. ^ , ^ . X cos tt 11 sin X - , d(u, v) Prob. 50. Given =o, — : — = o; to find ,, , . V cos y ^ sin ^' d{x, y) Prob. 51. Obtain the Hessian and the discriminant of: (a) the binary cubic; (b) the quaternary quadric; (c) the binary quartic. Prob. 52. Find the Hessian of ax^ -\-by^-{- cz^ + 2/yz + 2gzx + 2 hxy, and also that of the same function transformed by the substitutions. x=liX^-\-miyi + niZ% y=l2x'+m2y^+n2Z% z=lzx'-\-m2,y'-\-nzz'. INDEX. Alternant, 21. Arrays, square, 10, 11. rectangular, 33. Binomial, factors, removal of, 20. Cauchy, 7, 11, 27. Cauchy's method of expansion, 26. Cayley, 7. Co-factors, 21, 37. Columns and rows, 10, 16, 17. Composition of parallel lines, 19. Consistence of equations, 33. Covariants, 45, 50, 52. Cramer, 7. De L'Hospital, 7. Development of determinants, 23. Differentiation, 28. Discriminant, 46, 47. Eliminant, of linear systems, 35. Sylvester's dialytic, 38. Equations, linear systems of, 31, 40. Even permutations, 8. Gauss, 7. Hessian, 51. Homogeneous systems, 35. Implicit functions, Jacobian of, 50. Indirect functions, Jacobian of, 49. Interchange of elements, 9. of rows and columns, 16. Invariant, 46, 47. Inversions and permanences, 8, 9. Jacobi, 7. Jacobian, 48. Lagrange, 7. Laplace, 34. Leibnitz, 7, Linear systems of equations, 31, 33, 35. transformations, 40, 44. Lowering the order, 30. Matrix, 35. Minor, 21. Multiplication of determinants, 18, 40. of matrices, 43. theorem, 40. Negative permutations, 8, 10. Notation of determinants, 13. Odd permutations, 8. Order of determinant, 12. Parallel lines, 17. Permanences and inversions, 8, 9. Permutations, positive and negative, 8, 10. Polynomial elements, 18. Positive permutations, 8, 10. c. c a, c 56 Product of determinants, 40, 42. of rectangular arrays, 43. Quadric, 45, 51. Quantic, 45, 46, 47. Raising the order, 29. Reciprocal determinants, 44. Resultant, 34. Sarrus's rule, 14. INDEX Solution of linear systems, 31. Sylvester, 7, 38. Sylvester's method of elimination, 38. Transformations, linear, 44. unimodular, 45. Unimodular transformations, 45. Vandermonde, 7. 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Small Hospitals, Establishment and Maintenance; Suggestions for Hospital Architectxire: Plans for Small Hospital. 1 2 mo, i 25 HEBREW AND CHALDEE TEXT-BOOKS. Green's Elementary Hebrew Grammar i2mo, i 25 Hebrew Chrestomathy 8vo, 2 00 Gesenius's Hebrew and Chaldee Lexicon to the Old Testament Scriptures. (Tregelles.) Small 4to, half morocco, 5 00 Letteris's Hebrew Bible 8vo, 2 25 18 SHORT-TITLE CATALOGUE OF THE PUBLICATIONS OP JOHN WILEY & SONS, New York. LoHDOif: CHAPMAN & HALL, Limited, ARRANGED UNDER SUBJECTS. Descriptive circulars sent on application. Books marked with an asterisk (*) are sold at net prices only, a double asterisk (**) books sold under the rules of the American Publishers' Association at net prices subject to an extra charge for postage. All books are bound in cloth unless otherwise stated. AGRICULTURE. Armsby's Manual of Cattle-feeding lamo, Si 75 Principles of Animal Nutrition 8vo, 4 00 Budd and Hansen's American Horticultural Manual: Part I. Propagation, Culture, and Improvement i2mo, i 50 Part II. Systematic Pomology i2mo, i 50 Downing's Fruits and Fruit-trees of America 8vo, 5 00 Elliott's Engineering for Land Drainage i2mo, i 50 Practical Farm Drainage i2mo. i 00 Green's Principles of American Forestry i2mo, i 50 Grotenfelt's Principles of Modern Dairy Practice. (Well.) i2mo, 2 00 Kemp's Landscape Gardening i2mo, 2 50 Maynard's Landscape Gardening as Applied to*[ome Decoration i2mo, i 50* ^ McKay and Larsen's Principles and Practice of Butter-making 8vo, i 50 Sanderson's Insects Injurious to Staple Crops i2mo, i 50 Insects Injurious to Garden Crops. (In preparation.) Insects Injuring Fruits. (In preparation.) Stockbridge's Rocks and Soils 8vo, 2 50 Winton's Microscopy of Vegetable Foods 8vo, 7 50 Woll's Handbook for Farmers and Dairymen i6mOr i 50 ARCHITECTURE. Baldwin's Steam Heating for Buildings i2mo. 2 50 Bashore's Sanitation of a Country House i2mo, i 00 Ber;s's Buildings and Structures of American Railroads 4to, s 00 Birkmire s Planning and Construction of American Theatres 8vo, 3 00 Architectural Iron and Steel 8vo, 3 50 Compound Riveted Girders as Applied in Buildings 8vo, 2 00 Planning and Construction of High Office Buildings 8vo, 3 so Skeleton Construction in Buildings 8vo, 3 00 Brigg's Modern American School Buildings Svo. 4 00 Carpenter's Heating and Ventilating of Buildings Svo, 4 00 Freitag's Architectural Engineering Svo, 3 50 Fireproofing of Steel Buildings Svo, 2 50 French and Ives's Stereotomy. . , Svo, 2 50 1 I CO I 50 2 SO 75 2 oo 5 oo 5 oo 4 oo 4 oo 5 oc 7 5C 4 oo 3 oo 1 so 3 so 2 oo 3 oo 6 oo 6 so 5 oo 5 so 3 oo 4 oo I 25 I 00 Geriiaids Guide to Saiiltey XIo«se*inBpection i6mo, Theatre Fires and Panics i2mo. ♦Greene's Structural Mechanics . . . 8vo, Holly's Carpenters' and Joiners' Handbook i8mo, Johnson's Statics by Algebraic and Graphic Methods 8vo, Kidder's Architects' and Builders' Pocket-book. Rewritten Edition. i6mo, mor., Merrill's Stones for Building and Decoration Svo, Non-metallic Minerals : Their Occurrence and Uses Svo, Monckton's Stair-building ^ 4to, Patton's Practical Treatise on Foundations Svo, Peabody's Naval Architecture Svo, Richey's Handbook for Superintendents of Construction i6mo, mor., Sabin's Industrial and Artistic Technology of Paims and Varnish Svo, Siebert and Biggin's Modern Stone-cutting and Masonry Svo, Snow's Principal Species of Wood Svo, Sondericker's Grapnic Statics with Applications to Trusses, Beams, and Arches, Svo, Towne's Locks and Builders' Hardware iSmo, morocco, Wait's Engineering and Architectural Jurisprudence Svo, Sheep, Law of Operations Preliminary to Construction in Engineering and Archi- tectxire Svo, Sheep, Law of Contracts Svo, Wood's Rustless Coatings: Corrosion and Electrolysis of Iron and Steel. .Svo, Worcester and Atkinson's Small Hospitals, Establishment and Maintenance, Suggestions for Hospital Architecture, with Plans for a Small Hospital. i2mo, The World's Columbian Exposition of 1S93 Large 4to, ARMY AND NAVY. Bernadou's Smokeless Powder, Nitro-cellulose, and the Theory of the Cellulose Molecule i2mo, 2 50 * Bruff's Text-book Ordnance and Guftnery Svo, 6 00 Chase's Screw Propellers and Marine Propulsion Svo, 3 00 Cloke's Gunner's Examiner Svo, i 50 Craig's Azimuth 4to, 3 50 Crehore and Squier's Polarizing Photo-chronograph Svo. 3 00 * Davis's Elements of Law Svo, 2 50 * Treatise on the Military Law of United States Svo, 7 00 Sheep, 7 50 De Brack's Cavalry Outposts Duties. (Carr.) 24mo, morocco, 2 00 Dietz's Soldier's First Aid Handbook i6mo, morocco, i 25 * Dredge's Modern French Artillery 4to, half morocco, 15 00 Durand's Resistance and Propulsion of Ships Svo, s 00 * Dyer's Handbook of Light Artillery i2mo, 3 00 Eissler's Modern High Explosives Svo, 4 00 * Fiebeger's Text-book on Field Fortification Small Svo, 2 00 Hamilton's The Gunner's Catechism iSmo, i 00 * Hoff's Elementary Naval Tactics Svo, i 50 lagalls's Handbook of Problems in Direct Fire Svo, 4 00 * Ballistic Tables Svo, i 50 * Lyons's Treatise on Electromagnetic Phenomena. Vols. I. and II. .Svo, each, 6 00 * Mahan's Permanent Fortifications. (Mercur.) Svo, half morocco, 7 50 Manual for Courts-martial i6mo, morocco, i 50 * Mercur's Attack of Fortified Places i2mo 2 00 * Elements of the Art of War Svo, 4 00 2 14 DAY USE RSrURN TO DESK FROM WHICH BOSROWED LOAN DEPT. This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. l7Ju^65A^ jn^ -66 40^^9 LD 21A-60m-3,'65 (F2336sl0)476B General Library University of California Berkeley