Inverse of Plane Rotation Matrix

Theorem
Let $\mathbf R_1$ be the matrix associated with an anticlockwise rotation of the plane about the origin through an angle of $\alpha$.


 * $\mathbf {R_1} = \begin{bmatrix}

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

Let $\mathbf R_2$ be the matrix associated with a clockwise rotation of the plane about the origin through an angle of $\alpha$.


 * $\mathbf {R_2} = \begin{bmatrix}

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

Then:
 * $\mathbf {R_1}^{-1} = \mathbf {R_2}$


 * $\mathbf {R_2}^{-1} = \mathbf {R_1}$

Proof
The inverse matrix of an anticlockwise plane rotation matrix is:

The inverse matrix of a clockwise plane rotation matrix is:

Hence the result.