Systems, Matrices, and Determinants · Challenge problem

A rotation matrix $R(\theta)=\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}$ is the coefficient matrix of the system $R(\theta)\mathbf x=\mathbf b$. Which statement correctly explains why this system has exactly one solution for EVERY angle $\theta$ and EVERY vector $\mathbf b$?