Inverse of Plane Rotation Matrix

Theorem
Let $\mathbf R$ be the matrix associated with a rotation of the plane about the origin through an angle of $\alpha$:


 * $\mathbf R = \begin{bmatrix}

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

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


 * $\mathbf R^{-1} = \begin{bmatrix}

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

Proof
Let:
 * $\mathbf A = \begin{bmatrix}

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

Consider $\mathbf R \mathbf A$:

Now consider $\mathbf A \mathbf R$:

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