I think i've sort of got this. I just want someone to check what i've done.Suppose A and P are n x n matrices, and that P is invertible, and n is a positive integer. Show that $\displaystyle (P^{-1}AP)^n=P^{-1}A^n P$

$\displaystyle P^{-1}AP=\begin{pmatrix}

{\lambda_1}&{0}\\

{0}&{\lambda_2}

\end{pmatrix}$

Where $\displaystyle \lambda_1$ and $\displaystyle \lambda_2$ are eigenvalues.

Therefore $\displaystyle (P^{-1}AP)^n=\begin{pmatrix}

{\lambda_1}&{0}\\

{0}&{\lambda_2}

\end{pmatrix}^n=\begin{pmatrix}

{\lambda_1^n}&{0}\\

{0}&{\lambda_2^n}

\end{pmatrix}$

$\displaystyle P^{-1}A^nP=\begin{pmatrix}

{\lambda_1^n}&{0}\\

{0}&{\lambda_2^n}

\end{pmatrix}$

and since they are the same then the proof is complete.

Is this right?