Find matrice $A_{2 \times 2}$ such $A_{2 \times 2}\in \mathbb{R}$ such that $A^{30}=I$ I have the following question : Find $A_{2 \times 2}$ matrice $A_{2 \times 2}\in \mathbb{R}$ such that $A^{30}=I$ I tried to solve this problem using determinants using $|A^2|=|A|*|A|$, I think that method is could lead to the answer yet I don't seems to find $A$. Any ideas? Any help will be appreciated. EDIT : In case $A \neq I$ and the first time $A^k=I$ is for $k=30$

**Hint** : The rotation matrix $$A = \begin{bmatrix}\cos\theta & -\sin\theta \\\ \sin\theta & \cos\theta\end{bmatrix}$$ satisfies $$A^k = \begin{bmatrix}\cos k\theta & -\sin k\theta \\\ \sin k\theta & \cos k\theta\end{bmatrix}.$$

In other words, applying $k$ rotations by an angle of $\theta$ is the same as one rotation by an angle of $k\theta$.

Also, a rotation matrix is the identity if and only if the angle of rotation is a multiple of $360^{\circ}$.

Using these facts, can you think of an angle $\theta$ such that $A^{30} = I$ but $A^k \
eq I$ for $k = 1,2,\ldots,29$?