If $A = \begin{bmatrix}1 & 2 \\ 0 & 1\end{bmatrix}$, what is $A^5$?

Explanation

1. 01

   $A^2 = \begin{bmatrix}1 & 4 \\ 0 & 1\end{bmatrix}$ and $A^3 = \begin{bmatrix}1 & 6 \\ 0 & 1\end{bmatrix}$: each power adds 2 to the top-right element.

2. 02

   So $A^n = \begin{bmatrix} 1 & 2n \\ 0 & 1 \end{bmatrix}$, and $A^5 = \begin{bmatrix}1 & 10 \\ 0 & 1\end{bmatrix}$.