Complex Eigenvalues and Rotation-scaling Matrices Worksheet with Answer Key

What's on the worksheet

1. Each matrix is a rotation-scaling matrix.
   \begin{bmatrix} -2 & -2 \\ 2 & -2 \end{bmatrix}\begin{bmatrix} 0 & 3 \\ -3 & 0 \end{bmatrix}\begin{bmatrix} \sqrt{3} & -1 \\ 1 & \sqrt{3} \end{bmatrix}\begin{bmatrix} -5 & 12 \\ -12 & -5 \end{bmatrix}
2. Let A = \begin{bmatrix} -11 & -6 \\ 15 & 7 \end{bmatrix}.
3. Let A = \begin{bmatrix} -3 & -5 \\ 1 & 1 \end{bmatrix}.
4. Let A = \begin{bmatrix} 3 & -5 & 0 \\ 1 & -1 & 0 \\ 1 & -5 & 4 \end{bmatrix}.
5. Let A = \begin{bmatrix} 5 & -2 \\ 4 & 1 \end{bmatrix}, \qquad \mathbf{u} = \begin{bmatrix} 1 - i \\ 2 \end{bmatrix}, \qquad \mathbf{w} =…
6. Let A be a real 2 × 2 matrix with eigenvalues a ± bi, where b ≠ 0.