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?
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1 Answer
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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 $\mid A\mid = 0$.
