# Solution to the Rotation Matrix — Inverse

Find the inverse of the rotation matrix where $$\theta$$ is a fixed angle. Then use your result to solve the system $$x=a \cos \theta-b\sin\theta$$, $$y=a\sin\theta+b\cos\theta$$ for $$a$$ and $$b$$ in terms of $$x$$ and $$y$$.

I'm just really not sure on where to begin with this. Do I need to use a determinant because it is a 2x2 matrix?

$$\begin{pmatrix} x \\ y \end{pmatrix} = \begin{bmatrix} \cos \theta & - \sin\theta \\ \sin \theta & \cos\theta \end{bmatrix} \begin{pmatrix} a \\ b \end{pmatrix}$$
If the 2×2 matrix is a rotation, when you invert it you will get the inverse rotation. So you either do it the long way (with 2×2 matrix inversion) or the short way of negating $\theta$.
Hint: What is the opposite operation of rotating a vector by $\theta$ in the anti-clockwise direction?