ALGEBRAIC th
column be identical witb the i the determinant A vanishes identically ; hence if j be not equal to i, k, or r, ctij cioj a-sj 0 = S aw ci2k a-Ak Aij. alr a2r a3r Similarly, by putting one or more of the deleted rows or columns equal to rows or columns which are not deleted, we obtain, with Laplace, a number of identities between products of determinants of complementary orders. Multiplication.—From the theorem given above for the expansion of a determinant as a sum of products of pairs of corresponding determinants it will be plain that the product of A = (an, a22,.--ann) and D = (&n, £>22, bnn) may be written as a determinant of order 2n, viz.— 0 cc 0/2i cis ... an — 1 0 0 0 a12 a-22 a«32r • • • a««2 0—1 0 0 ®13 &23 3 i • ■ ■ n3 o 0—1 IABI •i d
(l^n dsn ... arm 0
0 0 “ IG DI 0 0 0 ... 0 £>12 £>13 £>ln for brevity. 0 0 0 ... 0 £>21 £>22 £>23 £>2n 0 0 0 ... 0 £>31 £>32 £>33 £>3n 0 0 0 ... 0 £>„i £>„2 &«3 £>,.. Multiply the l‘f, th2nd,...nth rows by £>n, £>12, ...£>i« respectively, and,h add to the n + row; by £>21, b-n,.■-bin, andrd add to the n + 2 row; by £>31, £>32, ...£>3n and add to the n + 2> row, &c. C then becomes ail£>ll + O-12&12 + ... + Ctinbin, >221 £>11 + a22b + ... + Ovnbin, ... anbi + an2bi2 +... + anvb
0^11 £>21 + &12£>22 + . • • + Clinbzn, ft21 £>21 + (l^byi + ... + Oinb-lni ... «nl£>21 + ®m2£>22 + . • • + anriJC!.n «ll£>31 + «12£>32 + • - + CliHb?in, «2i£>3i + K22&32 + • • • + Cl9nbzn, ... a ,ii£>31 + an2b32 + •. • + annb2n aibn + <2i2&m2 + ... + ainhnn, a2ibn + a^bni + • • • + a2„bn„, .. .anbn + an2bn2 + .,. + a„nbnn and all the elements of D become zero. Now by the expansion theorem the determinant becomes ( _ )l+2+34~• .+2»b .C = ( - l)«<2«+1>+«C = C. We thus obtain for the product a determinant of order n. AVeth may say that, in the resulting determinant, the element in the I row and kth column is obtained by multiplying the elements in th the kth row of the first determinant severally by the elements in the i row of the second, and has the expression akibn + aiaba + a^ba... + (iknbin, and we obtain other expressions by transforming either or both determinants so as to read by columns as they formerly did by rows. Remark.—In particular the square of a determinant is a determinant of the same order (£>n&22£>33---£>««) such that bik — bki; it is for this reason termed symmetrical. The Adjoint or Reciprocal Determinant arises from A = (<2ii<222ff3:s ...ann) by substituting for each element the corresponding minor Ait so as to form D —(A11A22A33... A^).. If we form the product A.D by the theorem for the multiplication of determinants we find that the element in the ith row and kth column of the product is ®fciAji + ttyteAia + ... + a/m^in, the value of which is zero when k is different from 1, whilst it has the value A when k = i. Hence the product determinant has the principal diagonal elements each equal to A and the remaining elements zero. Its value is therefore A" and we have the identity D. A = A" or D = A™-1. It can now be proved that the first minor of the adjoint determinant, say Brs, is equal to An~2ars. From the equations + ay2X/> + ff 13213 + ... = £1 , a2X + (122252 + (*23253 + • • • — £2 , ®3l25l +-(132252 +((33253 + • • • — £3 , we derive
and thence
FORMS
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In general it can be proved that any minor of order p of the adjoint is equal to the complementary of the corresponding minor of the original multiplied by the p - B* power of the original determinant. Theorem.—The adjoint determinant is the 71-1^ power of the original determinant. The adjoint determinant will be seen subsequently to present itself in the theory of linear equations and in the theory of linear transformation. Determinants of Special Forms.—It was observed above that the square of a determinant when expressed as a determinant of the same order is such that its elements have the property expressed by aik — au- Such determinants are called symmetrical. It is easy to see that the adjoint determinant is also symmetrical, viz., thsuch that A;* = AH, for the determinant got by suppressing the i row and kth column differs only by an interchange of rows and columns from that got by suppressing the kth row and ith column. If any symmetrical determinant vanish and be bordered as shown below '■12 "-22 "23 A-. Ao Ao it is a perfect square when considered as a function of A1; A2, A3. For since A= Aaw, with similar relations, we have a number of relations similar to A11A22 = Af2, and either Ars = + ,J(krrkss) or - v/(ArrA„) for all different values of r and s. Now the determinant has the value - {Af Au + A^A^ + A§A33 + 2A2A3A03 + 2A3A1A31 + 2AJA2AJ2} = - SA|Arr-22ArA1Ari in general, and hence by substitution + {Aj V/A11+A2V£A22+ ... +AnN/A„„}“. A skew symmetric determinant has arr=0 and ars= - a,r for all values of r and s. Such a determinant when of uneven degree vanishes, for if we multiply each row by -1 we multiply the determinant by (-I)'t= -1, and the effect of this is otherwise merely to transpose the determinant, so that it reads by rows as it formerly did by columns, an operation which we know leaves the determinant unaltered. Hence A=- A or A = 0. When a skew symmetric determinant is of even degree it is a perfect square. This theorem is due to Cayley, and reference may be made to Salmon’s Higher Algebra, 4th ed. Art. 39. In the case of the determinant of order 4 the square root is A12A34 - A13A24 + A24A23. A skew determinant is one which is skew symmetric in all respects, except that the elements of the leading diagonal are not all zero. Such a determinant is of importance in the theory of orthogonal substitution. In the theory of surfaces we transform from one set of three rectangular axes to another by the substitutions X = aa5 + by+ cz, Y = a'x + b'y + c'z, Z = a"x + b"y + c"z, where X2 +Y2 + Z2=a52 + 7/2 + s2. This relation implies six equations between the coefficients, so that only three of them are independent. Further we find x — aX + a’Y + a"Z, 7/ = £>X + £>'Y + £>"Z, z = cX + c’Y + c"Zy and the problem is to express the nine coefficients in terms of three independent quantities. In general in space of n dimensions we have n substitutions similar to Xi=a-^x-L+avlXr, + ... + alnxn, and we have to express the ti2 coefficients in terms of (n-l), independent quantities ; which must be possible, because X? + X| + ... +X2 =35f +x^ +a;2 + ... +a;2. Let there be 2?i equations X 1 = £>ll£l + £>12?2 + £>13^3 + • • • > 252 = £>2l£l + £>22^2 L £>23?3 + • • •,
A25i = An£i + A2i£2 + A3i£3 + ... , Ax-2 = Ai2£l + A22£2 + A32£3 + ... , Arcs = Ai3£i + A2s£2 + Assh + • • • ,
Nx=£>ii£i + £>2i£2 + £>3i?3 + • • •, x2 = £>12£1 + £>22^2 + £>32S3 + • • • j
A”-1^£1 — BiiAa:i + B12A252 + B13Aa53 + ... , A” £2 = B2iAaq + IL^Arcg + B23Aa53 + ... , A’,_1£3 = B3iAa;i + B32Acf2 + B33A253 + ...,
where £>rr = l and £>rj= -bsr for all values of r and s. There are then|7i(re-l) quantities £>rs. Let the determinant of the &s be A6 and B„, the minor corresponding to bn. We can eliminate the quantities £l, £2, ■ ■ -£n and obtain n relations AftX1 = (2B11- Aft)a51 + 2B21a52 + 2B3jX3 + ..., AftX2 = 2B j2^i + (2B22 — Aft)352 + 2B32353 + • • •,
and comparison of the first and third systems yields Brs-A”_2(r,.j.
