An Extension of Determinants

Extract

If the elements of a determinant of order n are given by

a_rs(r, s = 1, 2, ... n),

then the rule for determining the sign to be affixed to any term of the determinant is usually given in a form equivalent to the following arrange the n elements of the term with one set of suffixes in ascending order (i.e. 1, 2, ... n), and count the number of inversions, equal to i, say, in the other set. Then the correct sign is ( − )ⁱ.