<s>[INST] Proving that $A+A^2+A^3$ has eigenvalue $\lambda+\lambda^2+\lambda^3$ where $\lambda$ is an eigenvalue of $A$ 

I know that $$P= \begin{pmatrix} 4 & 5 & 1 \\ -9 & -6 & 0 \\ 8 & 4 & 2 \end{pmatrix}$$

I also know that 
$$D= \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix}$$
Where $a,b,c$ are eigenvalues of $A+A^2+A^3$

In this case the eigenvalues of : 
$$A+A^2+A^3= \lambda + \lambda^2 + \lambda^3$$
Where $\lambda$ is an eigenvalue of A. Can someone prove this statement for me.
 [/INST] We know that a matrix $M^n = PD^nP^{-1}$.  Then $A = PDP^{-1}$ so
$$
A + A^2 + A^3 = PDP^{-1} + PD^2P^{-1} + PD^3P^{-1} = P(D+D^2+D^3)P^{-1}
$$
You have already found $P$ so once you have $D$ you just add.
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