Determinants, Swapping Rows

Swapping Rows

Let M be any matrix and let S be M with two adjacent rows swapped. Compare det₂(M) and det₂(S). Let p be a permutation product in det(M). Let p use column 7 for the first row and column 9 for the second. Let q be the corresponding permutation product on S, with 7 and 9 swapped. Now p and q multiply exactly the same entries, but in a diffferent order. Since multiplication is commutative, order doesn't matter. The two products are equal. Actually there is a catch; we transposed 7 and 9, and that changed the parity of the permutation. Thus the corresponding products are opposite, and det(S) = -det(M).

Every time we swap adjacent rows, or interchange any two rows, or apply an odd permutation to the rows of a matrix, the determinant is negated.

Using this swap principle, we can generalize det₁(M). The new formula, det₃(M), is recursive, like det₁(M), but you can designate any row, not just the first row. Delete the i^th row and j^th column, as j runs from 1 to n. Compute the subdeterminants, multiply by M_i,j, and negate when i+j is odd. Does this give the right answer? It does when i = 1. Note that i+j is odd precisely when j is even (since i = 1). Thus det₁ and det₃ are exactly the same when i = 1. Proceed by induction on i.

Swap row i with row i+1 and apply det₃ to row i+1, remembering that det₃ was correct when the same row was in position i. The subdeterminants are the same, and the products are the same, except i+j is odd when it used to be even, and even when it use to be odd, since i has increased by 1. The determinant has been negated, at least according to det₃. However, we know the determinant of a matrix is negated when two rows are swapped, so det₃ is correct.