Past exam of the mathematical methods I course of the University of Oxford / 2014 resit / 3

The determinant is an alternating multilinear map of its rows, so exchanging two rows gives the negative of the original determinant: $\det A\mapsto -\det A$. This also holds when the determinant is zero.

The determinant is unchanged by transposition: $\boxed{\det (A^T)=\det A}$. In the Leibniz formula for determinants, replacing each permutation by its inverse interchanges the row and column roles without changing its sign of a permutation.

Scaling the whole matrix scales all $n$ rows. The multilinearity of the determinant therefore givesThis includes $c=0$ for $n\ge 1$.