# Comparing determinants of an odd-dimension matrix

If $\mathbf{A}$ is a square matrix with odd dimensions, can $|\mathbf{A}|=|-\mathbf{A}|$?

This is true if $\mathbf{A}$ is over the field $\mathbb{Z}_2$, but are there any other situations?

## 1 Answer

If $A$ is $k\times k$ with $k$ odd, then $\mid -A\mid = (-1)^k\mid A\mid = -\mid A\mid$, so unless $-1 = 1$ in the field, this is possible only if $\mid A\mid = 0$.