Inverse of Plane Reflection Matrix

Theorem
Let $\mathbf R$ be the matrix associated with a reflection in the plane.


 * $\mathbf R = \begin{bmatrix}

\cos 2\alpha & \sin 2\alpha \\ \sin 2\alpha & -\cos 2\alpha \end{bmatrix}$

Then its inverse matrix $\mathbf R^{-1}$ is itself.

Proof
Consider $\mathbf R \mathbf R$:

Hence, by the definition of the inverse matrix, $\mathbf R$ is the inverse matrix of $\mathbf R$.