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The question is,

Let $M$ be an $n\times n$ matrix with real entries such that $M^3=I$. Suppose that $Mv$$\neq v$, for any nonzero vector $v$. Then which of the following statements is/are true?

(a) $M$ has real eigenvalues.

(b) $M$+$M^{-1}$ has real eigenvalues.

(c) $n$ is divisible by 2.

(d) $n$ is divisible by 3.

Now by simply using Cayley-Hamilton theorem, I managed to show that option (a) does not hold, as well as $M$+$M^{-1}$ has an eigenvalue 2. But how to know whether n is divisible by 2 or 3 or both? Please help.

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For (c) observe that the eigenvalues of $M$ are $\alpha$ and $\bar\alpha$ for some particular non-real number $\alpha$. Also $M$ is diagonalisable. The dimensions of the $\alpha$-eigenspace and the $\bar\alpha$-eigenspace should be the same.

For (d), think about the minimum polynomial of $M$ and its degree.

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