EDIT: the original formula was not valid for block matrices. The formula below is now correct.
Exercise: If $$ M = \begin{bmatrix} A & 0 \\ B & C \end{bmatrix} \in \mathsf{M}_n(\mathbb{C}),$$ then $$ M^p = \begin{bmatrix} A^p & 0 \\ \sum_{k=0}^{p-1} C^k B A^{p - 1 - k} & C^p \end{bmatrix}, \forall p \in \mathbb{N}. $$
I've seen this formula dozens of times over the years (there is also an upper-triangular version), but I can't seem to place a reference for it.
Any references would be greatly appreciated (I am only looking for a reference, I realize that it follows from induction and the definition of matrix multiplication).