<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] The complex numbers are isomorphic to the set of matrices of the form
$$
a+bi \sim
\begin{pmatrix} a & -b \\ b & a \end{pmatrix}
=a I + b J,
\text{ where }
J=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix},
J^2=-I
$$
Your first matrix corresponds to $z=\sqrt3 +i$.
Now $\dfrac{iz}{2}=\dfrac{-1+\sqrt3}{2}$ is a third root of unity and so $\left(\dfrac{iz}{2}\right)^6=1$.
Therefore $z^6=-64$ and the corresponding matrix is $-64I$.
The second matrix can be written $A=I+N$ where $N^2=0$.
Therefore, $A^k=I+kN$ by the binomial theorem.
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