... I aibs'ca.. I which proves the theorem.
10. Multiplying each element of a given line of a determinant by a given factor multiplies the determinant by that factor; for each term of the expansion contains a single element from the given line. The common factor thus appears once and only once in each term of the expan sion, and the determinant is, therefore, multi plied by that factor.
In the same way it may be shown that a determinant having a line of zeros is equal to zero. It also follows that if the elements of any line have a common ratio to the corresponding elements of any parallel line the determinant vanishes.
11. If each eletnent of a line of a determinant be multiplied by a given factor and the product added to the corresponding element of any parallel line the value of the determinant will not be changed. This follows directly from 9 to 10. Thus auainiu ... al* amausans ... awn auau(au + mau) ... ails " ' atuam(ana + ..mars,) chin 12. The terms of I ai(n) I which contain the element ai are those found in the expansion of ae 0 0 .. 0 .
(a) au(n) For if, in forming any term, another element than ai be taken from the first column an ele ment zero must be taken from the first row, and the term vanishes. It may readily be shown that the determinant (a) is equal to ... an(n) 1, (b) .. an(K) which is therefore the aggregate of the terms of I ai(n) I, (n-1)! in number, which 'contain the element ai'.
The determinant factor of order (n-1) by which the element a,' is multiplied in (b) is called the cofactor of that element in I al(n) I. It may be obtained from the given determinant by deleting the first column and the first row.
The cofactor of any element may be found in the same manner after transposing this element to the leading position. But this transposition multiplies the determinant by the sign factor (-1 )"+'. Hence, to find the co factor of delete its row and its column and give the resultant determinant the /positive .n when , . s (even\ OC + ) i odd/ ' The cofactor thus obtained is represented by As('), the sign factor (-1)sts being intrin For example, the cofactors of the elements of the second row of I a,'a,"at" I are A. E".- I I, A 11:_=_Ialck,aa::::1, I" 13. The _aggregates of terms containing the elements . .. of the determinant I aiN I are, respectively, a,,' ,a,," . . . a,,(n)A(n).
Each of these n aggregates includes (n-1) I terms of I I, no one of which appears in any of the others. In all of them, then, there are n(n-1) ! or n!, different terms of the deter minant, which is the whole number. Hence la,(n)1=an'An'-Eas"An" -I- . . . + (1) Similarly, Ph(*) I =a, (4 ,1,00 + at(a)A 2(0 + . . . ± (2) Any determinant may, by means of either (1) or (2), be resolved into determinants of an order one lower and thus, since A w', . .. A or A,('), . . . An(') are themselves determinants, it may ultimately be expressed in terms of deter minants of the third or second order, which may readily be expanded (see 5).
14. If the kth and sth rows of I a,(s) I are identical the elements an, an" , . .. (1/40) in formula (1) may be replaced by ah' as", .. .ah09 respectively. But in this case the value of the determinant is zero. Hence, h ands being different indices, . . . ah(n)AK(K)= 0. (3) Likewise p and s being different, -Fat(>Y)A" . . n(9*.--- 0. (4) 15. The determinant of order (n-1) obtained by depleting the sth row and the sth column of = an(s) lis called the minor of the determi nant with respect to the element and is written 4(0). Obviously, by what precedes, A(s)=( 1 )1-z 4( o_ If two rows, the kth and Kul, and two col umns, the pth and sth, are deleted the result is written and is called a minor of the second order. Minors of lower orders may be obtained in a similar manner and expressed by a similar notation.
Any mth minor of a given determinant and the determinant of the in elements at the inter section of the rows and columns deleted in forming it are called, with respect to each other, complementary minors. The determi nant may be expressed in terms of products of pairs of complementary minors, a method of expansion due to Laplace. Formula: (1) and (2) are special cases of the method. Its general statement is somewhat complicated.
16. The principles thus far developed will now be applied to the solution of systems of simultaneous linear equations; the process which, as stated at the outset, led to the dis covery and investigation of determinants. As sume the system of three equations aix biy -I- co= Kti (i 1, 2, 3.) In the determinant I stints I let the elements be replaced by the equal quantities ap pearing in the first members of the given equa tions. The two determinants now in hand are equal to each other ; thus buy + I silnc, air ca, be, = slbaci ctsx thy + ca, b,, c. But the first member of this equation may be separated into the determinants (see 9 and 10) x I a.b.ca I, y Ib,b,a I, and s l cib.c. the second and third of which are, by 8, equal to zero. Hence z I aibicK = Ktbscs I; or, explicity (see 1).
n,b,c, x= ab a3bsci Similarly, by starting with the determinants I alkss3 I and I alt4s, I, respectively, the values of y and a are found to be I ci,K,c, I I a,b.K. I.
1171.1Wri I a I It will be noted that the values of the un - knowns have for a common denominator the determinant of the coefficients of the given equation; while the numerator is, in each case, obtained from the denominator by replacing the column of coefficients of the unknown in ques tion by the column of absolute terms. The method is applicable to linear systems of any order.
17. When the number of given equations is greater than the number of unknowns their con sistency obviously depends upon some definite relation among the known elements. Let aix + boo ---- Ki (1= 1, 2, 3) be such a redundant system.
Solving the first two of these equations gives (see 1) s I alKI l I I aos y--= 1