Ex. In the determinant (a_1 b_2 c_3 d_4 e_5) what sign is to be prefixed to the element a_1 b_2 c_3 d_4 e_5 ?
Here 3, 1, and 2 are inverted with respect to 4; 1 and 2 are inverted with respect to 3, and 2 is inverted with respect to 5; hence there are six inversions and the sign of the element is positive.
558. Determinant of Lower Order. If in Art. 554, each of the constituents b_1, c_1, \ldots k_1 is equal to zero, the determinant reduces to a_1A_1; in other words it is equal to the product of a_1, and a determinant of the (n -1)th order, and we easily infer the following general theorem.
If each of the constituents of the first row or column of a determinant is zero except the first, and if this constituent is equal to m, the determinant is equal to m times that determinant of lower order which is obtained by omitting the first column and first row.
Also since by suitable interchange of rows and columns any constituent can be brought into the first place, it follows that if any row or column has all its constituents except one equal to zero, the determinant can be immediately expressed as a determinant of lower order.
This is sometimes useful in the reduction and simplification of determinants.
Ex. Find the value of
30 11 20 38 6 3 0 9 11 -2 36 3 19 6 17 22
Diminish each constituent of the first column by twice the corresponding constituent in the second column, and each constituent of the fourth column by three times the corresponding constituent in the second column, and we obtain
8 0 20 5 0 3 0 0 15 -2 36 9 7 6 17 4
and since the second row has three zero constituents, this determinant
