Determinant of elementary permutation matrix

Solution 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 $-1$.

Solution 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 $\pm 1$.