<s>[INST] Matrix $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ to a large power Compute
$\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}^{99}$
What is the easier way to do this other than multiplying the entire thing out?
Thanks
 [/INST] Consider the case of a general $2 \times 2$ matrix times $\left( \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right)$:
$\left( \begin{matrix} a & b \\ c & d \end{matrix} \right) \times \left( \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right) = \left( \begin{matrix} a & a + b \\ c & c + d \end{matrix} \right) $.
Thus, we see that multiplying any $2 \times 2$ matrix by $\left( \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right)$ once leaves the entries in the first column unchanged and increments the entries in the second column by the corresponding entries from the first column.  Since our lower-left entry was zero and the upper-left entry was one, this has the effect of simply incrementing the upper-right entry by one but leaving the rest unchanged.
Therefore, since we begin with $\left( \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right)$ and multiply it by $\left( \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right)$ 98 times, we see that $\left( \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right)^{99} = \left( \begin{matrix} 1 & 99 \\ 0 & 1 \end{matrix} \right) $.
</s>