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Suppose $x$ is an eigenvector of $P$, and let $\lambda$ be corresponding eigenvalue, then

$$Px=\lambda x$$

Also

$$P^3x=\lambda^3 x$$.

We are given $P^3=P$, which means

$$P^3x=Px \Rightarrow \lambda^3 x = \lambda x \Rightarrow (\lambda^3-\lambda)x=0$$

Since $x$ is non-zero, $\lambda^3-\lambda = 0$. Solving it, we get $\lambda={0,1,-1}$
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