Cyclic Subgroups and General Linear Group Show that for a constant $\theta$ 
$$H = \bigg\{\left(\begin{array}{cc}
 cos n \theta & -sin n \theta \\
 sin n \theta & cos n \theta
\end{array} \right) \Bigg| \textit{n} \in \mathbb{Z} \bigg\}$$
is a cyclic subgroup of $GL(n, \mathbb{R}).$ Do you think $H$ is finite? 

HINT: $$\begin{pmatrix}\cos n\theta & -\sin n\theta \\ \sin n\theta & \cos n\theta\end{pmatrix}\begin{pmatrix}\cos m\theta & -\sin m\theta \\ \sin m\theta & \cos m\theta\end{pmatrix} = \begin{pmatrix}\cos (n+m)\theta & -\sin (n+m)\theta \\ \sin (n+m)\theta & \cos (n+m)\theta\end{pmatrix}.$$
You can conclude that $$\begin{pmatrix}\cos n\theta & -\sin n\theta \\ \sin n\theta & \cos n\theta\end{pmatrix} = \begin{pmatrix}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{pmatrix}^n.$$
