11/27 作业

Homework 15

1. $\begin{pmatrix} 7 & 6 & -13 \\ 11 & 9 & 17 \end{pmatrix}$ .

2. (1) $\begin{pmatrix} 1 & -5 & 4 \\ 0 & 1 & -6 \end{pmatrix}$ .

   (2) $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ .

   (3) $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ .

3. a. $\begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ .

   b. $\begin{pmatrix} \cos \theta & 0 & \sin \theta \\ 0 & 1 & 0 \\ -\sin \theta & 0 & \cos\theta \end{pmatrix}$ .

4. a. $D$ .

   b. $E$ .

   c. $C$ .

   d. $A$ .

   e. $F$ .

5. $\begin{pmatrix} 1 & 0 \\ 2 & 0 \end{pmatrix}$ .

6. Yes, $\begin{pmatrix} 0 & -\mathbf v_3 & \mathbf v_2 \\ \mathbf v_3 & 0 & -\mathbf v_1 \\ -\mathbf v_2 & \mathbf v_1 & 0 \end{pmatrix}$ .

7. Yes, $\begin{pmatrix} \mathbf v_1 & \cdots & \mathbf v_m \end{pmatrix}$ .

8. $A^{-1} = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix}$ , rotation by $-\theta$ .

9. $\begin{pmatrix} 8 & -3 \\ -1 & 1 \end{pmatrix}$ .

10. $A = BS^{-1}$ .