<s>[INST] How to use powers on matrices In the questions compute $\begin{pmatrix} \sqrt{3} & -1 \\ 1 & \sqrt{3} \end{pmatrix}^6$ and $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}^{99}$, how would you solve these?
 [/INST] $\begin{pmatrix} \sqrt{3} & -1 \\ 1 & \sqrt{3} \end{pmatrix}= 2 \begin{pmatrix} \cos \frac {\pi}{6} & -\sin\frac {\pi}{6}  \\ \sin\frac {\pi}{6} & \cos \frac {\pi}{6} \end{pmatrix}$
$\begin{pmatrix} \sqrt{3} & -1 \\ 1 & \sqrt{3} \end{pmatrix}^6= 2^6 \begin{pmatrix} \cos \frac {\pi}{6} & -\sin\frac {\pi}{6}  \\ \sin\frac {\pi}{6} & \cos \frac {\pi}{6} \end{pmatrix}^6$
It is worth the exercise to see what happens when you multiply matrices that can be put into this form.
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