### Instruction: Let $\mathbf{R}$ be the matrix for reflecting over the vector $\begin{pmatrix} 3 \\ 1 \end{pmatrix}.$  Find $\mathbf{R}^2.$.### Response: I know that reflecting over a vector twice means returning to the original position, so $\mathbf{R}^2$ should be the identity matrix $\mathbf{I}.$
To check this, I can use the fact that $\mathbf{R}$ is an orthogonal matrix, which means that $\mathbf{R}^T\mathbf{R} = \mathbf{I}.$
Therefore, $\mathbf{R}^2 = \mathbf{R}^T\mathbf{R} = \mathbf{I}.$