Determinant of elementary permutation matrix

Question

Why is the determinant of an elementary permutation matrix equal to $-1$?

I am brand new to determinants and I've tried expanding it and using cofactor expansion, but it's messy and complicated. I would prefer if someone could show me using expansion, but alternative methods are welcome. Thanks.

Answer 1

Not every permutation matrix has determinant $-1$, but the elementary matrices which are permutation matrices (corresponding to interchanges of two rows) have determinant $-1$. The easy way to see this is that (1) the identity matrix has determinant $1$, and (2) interchanging two rows or columns of a matrix multiplies its determinant by $-1$.

Answer 2

Recall that if we interchange two rows (or two coulumns) of a matrix $A$ then its determinant change the sign.

The permutation matrix is obtained from the identity matrix (with determinant $1$) by interchanging their rows so its determinant is $\pm 1$.