Exercise 6.8.22

See the paragraph after Theorem 6.35.

1.

Take

$$\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill -3\hfill & \hfill 1\hfill \end{array}\right)A\left(\begin{array}{cc}\hfill 1\hfill & \hfill -3\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -7\hfill \end{array}\right).$$

2.

Take

$$\left(\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill -\frac{1}{2}\hfill & \hfill \frac{1}{2}\hfill \end{array}\right)A\left(\begin{array}{cc}\hfill 1\hfill & \hfill -\frac{1}{2}\hfill \\ \hfill 1\hfill & \hfill \frac{1}{2}\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill 2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -\frac{1}{2}\hfill \end{array}\right).$$

3.

Take

$$\left(\begin{array}{ccc}\hfill 1\hfill & \hfill -\frac{1}{4}\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right)A\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -\frac{1}{4}\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right)=\left(\begin{array}{ccc}\hfill \frac{19}{4}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 4\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill \end{array}\right).$$