Calculating the Eigenvectors and Eigenvalues of this Matrix Polynomial

Question

For the matrix $$ A=\begin{pmatrix} 1 & 1 & 2 \\ 0 & -2 & 0 \\ 0 & 2 & 3 \end{pmatrix} $$

How are the eigenvalues and eigenvectors of the following matrices calculated?

$$ 3A^4-2A^3+A-2I $$

and

$$ A^{2n}-I $$

(n is a natural number)

Answer 1

HINT: If $Ax=\lambda x$, then $$(c_nA^n+\cdots +c_1A+c_0I)x=c_n\lambda^nx+\cdots +c_1\lambda x+c_0x=(c_n\lambda^n+\cdots +c_1\lambda+c_0)x$$

Answer 2

Hint:

If you have a matrix A with eigenvalues $\lambda_i$. and eigenvectors $v_i$, then the eigenvalues of $A^2$ will be $\lambda_i^2$ and the eigenvectors will still be $v_i$.

Also, the eigenvalues of $A+A^n$= the sum of the eigenvalues of both of those matrices.

We can first diagonalize $A$ as $A=PDP^{-1}$ where $P$ is the eigenvectors in it's columns and $D$ is the eigenvalues in the diagonal.

$$3A^4=3PD^4P^{-1}\\-2A^3=-2PD^3P^{-1}\\A=PDP^{-1}\\3A^4-2A^3+A=3PD^4P^{-1}-2PD^3P^{-1}+PDP^{-1}=P(3D^4-2D^3+D)P^{-1}$$

That is as far as i can get to be honest. I don't know how the $I$ plays in