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Suppose we have a $10 \times 10$ matrix $A$ which has $0$'s on the main diagonal (so that the trace of $A$ is $0$). Also suppose that $A^2=I$. How can we find a determinant of $A+2I$?

Based on my previous question, I tried to find eigenvalues, as in find $\lambda$ so that $A-\lambda I$ is not invertible. I have trouble again doing this, given the hypotheses here that are different than those of my previous question. How can I proceed in finding eigenvalues so I can construct a spectrum and find the determinant?

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user321401
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    $A^2 = I$ means that the eigenvalues are either plus or minus 1, as, $x = A^2 x = \lambda^2 x$ for any eigenvalue of $A$. Since the trace of $A$ is zero, the sum of the eigenvalues is zero, and thus the eigenvalues are 1,1,1,1,1,-1,-1,-1,-1,-1, ie, 5 1's and 5 -1's. – James Kilbane Mar 15 '16 at 10:27

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