<s>[INST] How to find $A^{100}$ where $A$ is a matrix with trigonometric entries? 
How to find $A^{100}$ where $A=\left(\begin{matrix}\cos \theta &  -\sin \theta\\  \sin \theta & \cos \theta\end{matrix}\right)$. ? (Given$\theta=\frac{2\pi}{7}$)

I am a highschool student and new in linear algebra and seeking for only problem solving strategies. I found somewhere the trick to solve this kind of problem is find characteristic equation and then for some of the cases it is possible to find any power of matrix just by exponentiation of both sides. But here this is not applicable.
The characteristics equation found to be $\lambda^2-2\lambda \cos \theta+1=0$
I
how to proceed now ?
 [/INST] From the characteristic equation,
$$\begin{align*}
\lambda^2 - 2\lambda \cos \theta + 1 &= 0\\
\lambda &= \frac{2\cos \theta\pm \sqrt{4\cos^2\theta-4}}{2}\\
&= \cos\theta \pm i\sin\theta
\end{align*}$$
For $\lambda_0 = \cos\theta + i\sin\theta$,
$$\begin{align*}
\pmatrix{\cos \theta-\lambda_0 &-\sin\theta\\\sin\theta &\cos\theta-\lambda_0}\mathbf v &= 0\\
\pmatrix{-i\sin\theta &-\sin\theta\\\sin\theta &-i\sin\theta}\mathbf v &= 0\\
\pmatrix{-i&-1\\1&-i}\mathbf v &= 0\\
\mathbf v &= k\pmatrix{i\\1}\\
A\pmatrix{i\\1} &= (\cos\theta + i\sin\theta)\pmatrix{i\\1} \tag 1
\end{align*}$$
Similarly, for $\lambda_1 = \cos \theta - i\sin\theta$,
$$A\pmatrix{1\\i} = (\cos\theta - i\sin\theta)\pmatrix{1\\i}\tag2$$
$(1)\pmatrix{1 &0} + (2)\pmatrix{0&1}$,
$$\begin{align*}
A\pmatrix{i\\1}\pmatrix{1 &0} + A\pmatrix{1\\i}\pmatrix{0&1} &= \lambda_0\pmatrix{i\\1}\pmatrix{1 &0} + \lambda_1\pmatrix{1\\i}\pmatrix{0&1}\\
&= \pmatrix{i\\1}\pmatrix{\lambda_0 &0} + \pmatrix{1\\i}\pmatrix{0&\lambda_1}\\
A\pmatrix{i&1\\1&i} &= \pmatrix{i&1\\1&i}\pmatrix{\lambda_0 &0\\0&0} + \pmatrix{i&1\\1&i}\pmatrix{0 &0\\0&\lambda_1}\\
&= \pmatrix{i&1\\1&i}\pmatrix{\lambda_0 &0\\0&\lambda_1}\\
A &= \pmatrix{i&1\\1&i}\pmatrix{\lambda_0 &0\\0&\lambda_1}\pmatrix{i&1\\1&i}^{-1}\\
A^{n} &= \pmatrix{i&1\\1&i}\pmatrix{\lambda_0 &0\\0&\lambda_1}^{n}\pmatrix{i&1\\1&i}^{-1}\quad(n\in\mathbb N)\\
&= \pmatrix{i&1\\1&i}\pmatrix{\lambda_0^{n} &0\\0&\lambda_1^{n}}\pmatrix{i&1\\1&i}^{-1}\\
&=\frac1{-2} \pmatrix{i&1\\1&i}\pmatrix{\lambda_0^{n} &0\\0&\lambda_1^{n}}\pmatrix{i&-1\\-1&i}\\
&= \frac1{-2} \pmatrix{i&1\\1&i}\pmatrix{\cos n\theta + i\sin n\theta &0\\0&\cos n \theta - i\sin n\theta}\pmatrix{i&-1\\-1&i}\\
&= \frac1{-2} \pmatrix{i\cos n\theta -\sin n\theta &\cos n\theta - i\sin n\theta\\\cos n\theta + i\sin n\theta &i\cos n\theta +\sin n\theta}\pmatrix{i&-1\\-1&i}\\
&= \frac1{-2} \pmatrix{-2\cos n\theta&2\sin n\theta\\-2\sin n\theta &-2\cos n\theta}\\
&= \pmatrix{\cos n\theta&-\sin n\theta\\\sin n\theta &\cos n\theta}\\
\end{align*}$$
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