Determinants, The Transpose of M

The Transpose of M

Let S be the transpose of M, and compare det₂(M) and det₂(S). If p is a permutation applied to M, let q be the inverse permutation applied to S. Note that p and q have the same parity (both even or both odd). Verify that p and q multiply exactly the same entries from M and S, but in a different order. The products are identical, hence det(M) = det(S). Taking the transpose of a matrix does not change its determinant.

This leads to another definition of determinant, which we will call det₄(M). Delete a fixed column and the j^th row, as j runs from 1 to n. Multiply subdeterminants by entries in the deleted column, and add up the products. This is just like det₃, but we're moving down a column.

Let S be the transpose of M, and compute det₄(M) and det₃(S). Fix column i in M and row i in S, giving the same subdeterminants and the same entries. Det₄ gives exactly the same formula as det₃. Thus det₄ gives the determinant of the transpose of M, which happens to be the determinant of M.