### Instruction: Let $\mathbf{D}$ be a matrix representing a dilation with scale factor $k > 0,$ and let $\mathbf{R}$ be a matrix representing a rotation about the origin by an angle of $\theta$ counter-clockwise.  If
\[\mathbf{R} \mathbf{D} = \begin{pmatrix} 8 & -4 \\ 4 & 8 \end{pmatrix},\]then find $\tan \theta.$.### Response: I know that a dilation matrix $\mathbf{D}$ has the form \[\mathbf{D} = \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix},\]where $k$ is the scale factor.
I also know that a rotation matrix $\mathbf{R}$ has the form \[\mathbf{R} = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix},\]where $\theta$ is the angle of rotation counter-clockwise.
Multiplying these two matrices, I get \[\mathbf{R} \mathbf{D} = \begin{pmatrix} k \cos \theta & -k \sin \theta \\ k \sin \theta & k \cos \theta \end{pmatrix}.\]
Comparing this to the given matrix, I see that \[k \cos \theta = 8 \quad \text{and} \quad k \sin \theta = 4.\]