Powers of a $2 \times 2$ matrix Let $A$ be a $2 \times 2$ matrix such that $$A = \begin{pmatrix} \sin\frac{\pi}{18} & -\sin\frac{4\pi}{9} \\ \sin\frac{4\pi}{9} & \sin\frac{\pi}{18} \end{pmatrix}$$ Find the smallest number $n \in N$ such that $A^n = I$, where $I$ is the identity matrix of order $2$.

Using the identity $\sin\theta=\cos(\frac{\pi}{2}-\theta)$, you get $\sin\frac{\pi}{18}=\cos\frac{4\pi}{9}$.
Substitute this in and you get your standard rotation matrix:
$\begin{pmatrix}
\cos\frac{4}{9}\pi & -\sin\frac{4}{9}\pi \\ 
\sin\frac{4}{9}\pi  & \cos\frac{4}{9}\pi 
\end{pmatrix}$
This is equivalent to a counterclockwise rotation of $\frac{4}{9}\pi$. How many times do you need to apply this rotation so that you are back to where you started?
As $4$ and $9$ are coprime, you will need to do it $9$ times. Hence, $n=9$.
