Show that $p(A) = \begin{bmatrix} p(A_{11})&(Mess)\\ 0&p(A_{22})\\ \end{bmatrix}$ for any polynomial $p(x)$. (See problem for full question.) Important Note: This is a homework problem.
The full question is as follows:
If
$$A = \begin{bmatrix} A_{11}&A_{12}\\ 0&A_{22}\\ \end{bmatrix}$$
where $A_{11}$ and $A_{22}$ are square, show that
$$p(A) = \begin{bmatrix} p(A_{11})&(Mess)\\ 0&p(A_{22})\\ \end{bmatrix}$$
for any polynomial p(x).
Honestly I'm at a complete loss as to how I should even begin showing this is true. I know how to apply a polynomial to a matrix but that's about it. The only thing that comes to mind is that it might have something to do with diagonalization.
Thank you for any help in getting started with this.

Hint: Show the claim first for $p(x) = x^2$, then for $p(x) = x^n$, then for $p(x) = ax^n$, then in general.
