# Eigenvalues of sum of a matrix and its inverse

Let $$M$$ be a $$n\times n$$ matrix such that $$M^3=I$$.Suppose that $$Mv \neq v$$ for any non zero vector $$v$$.Then which of the following is\are true?

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

The solution i tried is

Here it is given that $$M^3=I$$ so from here it is confirm that its minimal polynomial can be from $$(x-1),(x^2+x+1) \;or\; (x-1)(x^2+x+1)$$, but according to given condition $$(x-1)\; and \;(x-1)\;(x^2+x+1)$$ can't be minimal polynomial.So the only possibility is $$(x^2+x+1)$$

From above it is confirmed that roots are complex roots ,but still i have no idea how to prove $$M+M^{-1}$$ has real eigenvalue

If $$x^{2}+x+1$$ is the minimal polynomial of $$M$$ then $$M^{2}+M+I=0$$. Also $$M^{3}=I$$ implies $$M^{-1}=M^{2}$$. Hence $$M^{-1}+M=M^{2}+M=-I$$.
Let $$\lambda$$ be an eigenvalue of $$M$$. Then, since $$M^3=\operatorname{Id}$$, $$\lambda^3=1$$. But\begin{align}\lambda^3=1&\iff(\lambda-1)(\lambda^2+\lambda+1)=0\\&\iff\lambda^2+\lambda+1=0,\end{align}since $$\lambda\neq1$$. There are only two numbers $$\lambda$$ such that $$\lambda^2+\lambda+1=0$$: $$-\frac12\pm\frac{\sqrt3}2i$$. In each case, $$\frac1\lambda=\overline\lambda$$. If $$v$$ is an eigenvector corresponding to the eigenvalue $$\lambda$$, then$$(M+M^{-1}).v=\lambda v+\frac1\lambda v=\left(\lambda+\overline\lambda\right)v=2\operatorname{Re}(\lambda)v=-v.$$