Find all eigenvalues of a matrix that holds $A = A^{5}$

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I need to find all eigenvalues of a matrix that holds $A = A^{5}$. I have no idea where to start. Any directions please?

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2 Answers

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Hint. Any eigenvalue $\lambda$ of $A$ will solve the equation $\lambda^5=\lambda$ because $$0=(A^{5}-A)v=(\lambda^5-\lambda)v$$ where $v$ is an eigenvector with respect to $\lambda$.

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Hint:

$$ \lambda^5-\lambda=0 \quad \iff \quad \lambda(\lambda-1)(\lambda+1)(\lambda^2+1)=0 $$

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