Exercise 5.3.9

Use the same method as that in Exercise 5.3.2. Or we may use the result of 5.20 for the case of a regular matrix.

1.

The limit is $\left(\begin{array}{ccc}\hfill \frac{1}{3}\hfill & \hfill \frac{1}{3}\hfill & \hfill \frac{1}{3}\hfill \\ \hfill \frac{1}{3}\hfill & \hfill \frac{1}{3}\hfill & \hfill \frac{1}{3}\hfill \\ \hfill \frac{1}{3}\hfill & \hfill \frac{1}{3}\hfill & \hfill \frac{1}{3}\hfill \end{array}\right)$.

2.

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

3.

The limit does not exist since the second and the third columns interchange each time.

4.

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

5.

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

6.

The limit is $\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{2}{5}\hfill & \hfill \frac{2}{5}\hfill \\ \hfill 0\hfill & \hfill \frac{3}{5}\hfill & \hfill \frac{3}{5}\hfill \end{array}\right)$.

7.

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

8.

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