Find the eigenvalues of $A^{3}$

Question

Given the matrix \begin{bmatrix} 4 & 0 & 1 \\ -2 & 1 & 0 \\ -2 & 0 & 1 \end{bmatrix}

I have found the eigenvalues to be $\lambda_{1} = 1$, $\lambda_{2} = 2$, $\lambda_{3} = 3$. How do I find the eigenvalues of $A^{3}$?

Answer 1

Hint: for an eigenvector $x$ of A

$A^3 x=AAAx=AA\lambda x =\dots = \lambda^3 x$.