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ALGEBRAIC FORMS theory of linear transformation. The matrix consists of n rows value of the determinant is unchanged. In general we can prove and n columns. Each row as well as each column supplies one in the same way the— Theorem.—The value of a determinant is unchanged if we and only one element to each member of the determinant. Consider ition of the definition of the determinant shows that the add to the elements of any row or column the corresponding value is unaltered when the suffixes in each element are transposed. elements of the other rows or other columns respectively each Theorem.—If the determinant is transformed so as to read by multiplied by an arbitrary magnitude, such magnitude remaincolumns as it formerly did by rows its value is unchanged. The ing constant in respect of the elements in a particular row or a leading mzmber of the determinant is ... ann, and corre- particular column. Observation.—Every factor common to all the elements of a sponds to the principal diagonal of the matrix. row or of a column is obviously a factor of the determinant, and We write frequently may be taken outside the determinant brackets. A = 2 + ^11^22^33' • -Unn = (®jl^22^33' • a2 /32 - cr Ex. gr. a2 /32 If the first two columns of the determinant be transposed the r — or y - a —a expression for the determinant becomes 2( - Ya^a^azy■ ■ .anv, viz., !-a 7—a | 1 1 1 0 0 a and /3 are transposed, and it is clear that the number of trans(4-a 74-a _ -y y + a positions necessary to convert the permutation /3ay...v of the -a)(7-a) _ (£ - y)(y ~ a) 0 1 1 1 second suffixes to the natural order is changed by unity. Hence the transposition of columns merely changes the sign of the = (/3-a)(7-a)(£- 7). determinant. Similarly it is shown that the transposition of any 0A The minor Aik is oa two columns or of any two rows merely changes the sign of the 5—, and is itself a determinant of order n-. ik determinant. Theorem.—Interchange of any two rows or of any two columns We may therefore differentiate again in regard to any element ars where r^i, s^k ; we will thus obtain a minor of Aik, which is merely changes the sign of the determinant. Corollary.—any two rows or any two columns of a deterdA1ik- =? 02A a minor also of A of order ■2. It will be Art—— minant be identical the value of the determinant is zero. 0((?*s 0((; ArC'U?-, Minors of a Determinant.—From the value of A we may separate and will be obtained by erasing from the determinant Aik the those members which contain a particular element a.;* as a factor, row and column containing the element a
this was originally and write the portion aik Aik ; Aik, the cofactor of aik, is called a the rth row and sth column of A ; the rth rsrow of A is the rth or minor of order w - 1 of the determinant. th th Now anAn = 1,±alla22a33... ann, wherein au is not to be (r - l) row of Aik according as rfi and the s column of A is changed, but the second suffixes in the product a22a33...ann the sth or s - Vh column of Aik according as s < k. Hence, if Th assume all permutations, the number of transpositions necessary denote the number of transpositions necessary to bring the sucdetermining the sign to be affixed to the member. ri into ascending order of magnitude, the sign to be Hence anAn = <zu2 + a22<x33 , where the cofactor of an is cession attached to thethdeterminant arrived at by erasing the ith and rth th clearly the determinant obtained by erasing the first row and the rows and the k and s columns from A in order produce Aik will first column.. rs be -1 raised to the power of Tri 4- T*s + i + k + r + s. ^22 ®33 ••• Similarly proceeding to the minors of order n-'d, we find that Hence Au= a32 - a3n Ai is obtained from A by k = — Aij; = sr-^—A | #n2 ®„3 ... ann I darsdat. ik = ^—^—A ’datl 0 a>i/- dars?ja ^ Similarly Aik, the cofactor of aik, is shown to he the product of the it'h, rth, tth rows, the kth, sth, nth columns, and multiplying (- )i+k andth the determinant obtained by erasing from A the itr> erasing the resulting by -1 raised to the power Ttri 4- T^ row and k column. No member of a determinant can involve + i + k + r + s +determinant t + u and the general law is clear. more than one element from the first row. Hence we have the Corresponding Minors.—In obtaining the minor Aik in the development rs A — Aqi-A.il q- ^22^10 -I oq3A|3 4-... 4- a A , form of a determinant we erased certain rows and columns, and proceeding according to the elements of the first row and the we would have erased in an exactly similar manner had we been forming the determinant associated with Als, since the deleting corresponding minors. Similarly we have a development proceeding according to the lines intersect in two pairs of points. In therk latter case the sign elements contained in any row or in any column, viz., is determined by -1 raised to the same power as before, with the exception that TMx4 replaces Tmk ; Imt if one of these numbers be A = a*! Aji + aviAa + al3Aiz +... + ®inA('„ even the other must be uneven ; hence A — «iitAla 4- a-2kAok 4- azkAzk 4-... + ankAnk Aik— A;* This theory enables the evaluation of a determinant by successive reduction of the orders of the determinants involved. Moreover Ex. gr. 10 3 2 1I 6 -0 2 6 Oikars-k ik. -f a i/irkA I.— " A; 2 16 0-5 I 3 0-3 rs rk ark r 0-5 3 -6 -5 | 4-3.2 -5 | -3.1 | 0 | where the determinant factor is given by the four points in which = 34-30-30-0 = 3. the deleting lines intersect. This determinant and that associated with Aik are termed corresponding determinants. Similarly p Since the determinant rs ttoi 022 023 ®2n lines of deletion intersecting in p2 points yield corresponding aa3 (122 ®23 ••• ®2n of orders p and n-p respectively. Recalling the having two identical rows, determinants 31 ((32 Ozz ... a3n formula A = UnAji 4- U-12A12 4- U13A13 4-... + ainAin, am an2 anz ... ann vanishes identically ; we have by development according to the it will be seen that ak and Au. involve corresponding determinelements of the first row ants. Since Alft is a determinant we similarly obtain a 2iAn + a22A12 + a23A13 4-... + a2nA
= 0 ;
Au=((21 Au 4-... + a2>&-iAi,*; 4- ^.ft-i-iA] ,* + ...+ o^nAi.*, and, in general, since 21 2,*-l 2,*+l 2,n (((1 Ajith4" 0^2 Ai2th"I OftAiz 4“ ... + OinAin — A, and thence if we suppose the i and k rows identical A = 2<2i/X2*Ak where i%k i,k 2k akAi 4- ak2Ai2 + ak3Aiz + • • • + aknAin = 0 (k<i) ; and as before and proceeding by columns instead of rows, A = 2 Mu a2i Au i>k, «iiAi* 4- a^Azii + a3iAzk 4-... 4- aniAnk = 0 (& < i) i,k ttlk Ct2k 2k identical relations always satisfied by these minors. an important expansion of A. If in the first relation of (A) we write ais = biS +Cis + diS+... we Similarly find that 2ai4Aij, = 26i,A;J4-2ci(Ai,4-2^isAij4-... so that A breaks an (% ozi up into a sum of determinants, and we also obtain a theorem for A = 2 aifc a2k azk Ah i>k> r, the addition of determinants which have n — 1 rows in common. alr a2r a3r 2k If we multiply the elements of the second row by an arbitrary magnitude X, and add to the corresponding elements of the first and the general theorem is manifest, and yields a developmentth row, A becomes 2aisAis4-X2a2sAis = 2aisAis = A, showing that the in a sum of products of corresponding determinants. If the j
