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If we have matrix $A\in Mat_{n}(\mathbb{R})$ with property $A^{2}=-I$. How to show n must be even?

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The shortest way is to take the determinant of both sides. –  Qiaochu Yuan Oct 26 '11 at 3:23
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Just take the determinant on both sides. You get $\det (A)^2 = (-1)^n$.

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