Now a11A11= alla22a33...ann, wherein all is not to be changed, but the second suffixes in the product a 22 a 33 ...a nn assume all permutations, the number of transpositions necessary determining the sign to be affixed to the member.
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= (a ll, a22, ï¿½ï¿½ï¿½ ann) and D = (b21, b 22, b nn) may be written as a determinant of order 2n, viz.
ï¿½ï¿½ a n1 - 1 a12 a 22 a32 ï¿½ï¿½ï¿½ an2 0 a13 a 23 a33 ï¿½ï¿½ï¿½ an3 0 a3n ï¿½ï¿½ ï¿½ a nn 0 0 0 ...
anibll +an2b12+ï¿½ ï¿½ï¿½ +annbin a11b21+a12b22+ï¿½ï¿½ï¿½ +alnb2n, a21b21+a22b22+ï¿½ï¿½ï¿½ +a2nb2n, ï¿½ ï¿½ ï¿½ ani b21 + a n2 b 22 + ï¿½ ï¿½ ï¿½ +annb2n alib31+a12b32+ï¿½ï¿½ï¿½+ainb3n, a21b31+a22b32+ï¿½ï¿½ï¿½+a2nb3n, .ï¿½.a n lb 31 + a n2 b 32+ ï¿½ï¿½ï¿½ +annb2n a ll b nl + a 12 b n2+ ï¿½ï¿½ï¿½ + a ln b nn, a21bn1+a22bn2+ï¿½-ï¿½+a2nbnn, ï¿½ ï¿½ ï¿½ ani b nl + a n2 b n2 +ï¿½ ï¿½ ï¿½ +annbnn and all the elements of D become zero.
In particular the square of a determinant is a deter minant of the same order (b 11 b 22 b 33 ...b nn) such that bik = b ki; it is for this reason termed symmetrical.