Exercise 5.3.2

Diagonalize those matrices and use the fact of the Corollary after Theorem 5.12. Actually the eigenvalue will not tend to zero if and only if the eigenvalue is $1$.

1.

The limit is the zero matrix.

2.

The limit is $\left(\begin{array}{cc}\hfill -0.5\hfill & \hfill 0.5\hfill \\ \hfill -1.5\hfill & \hfill 1.5\hfill \end{array}\right)$.

3.

The limit is $\left(\begin{array}{cc}\hfill \frac{7}{13}\hfill & \hfill \frac{7}{13}\hfill \\ \hfill \frac{6}{13}\hfill & \hfill \frac{6}{13}\hfill \end{array}\right)$.

4.

The limit is the zero matrix.

5.

The limit does not exist.

6.

The limit is $\left(\begin{array}{cc}\hfill 3\hfill & \hfill -1\hfill \\ \hfill 6\hfill & \hfill -2\hfill \end{array}\right)$.

7.

The limit is $\left(\begin{array}{ccc}\hfill -1\hfill & \hfill 0\hfill & \hfill -1\hfill \\ \hfill -4\hfill & \hfill 1\hfill & \hfill -2\hfill \\ \hfill 2\hfill & \hfill 0\hfill & \hfill 2\hfill \end{array}\right)$.

8.

The limit is $\left(\begin{array}{ccc}\hfill -2\hfill & \hfill -3\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 6\hfill & \hfill 9\hfill & \hfill 3\hfill \end{array}\right)$.

9.

The limit does not exist.

10.

The limit does not exist.