Chapter 8, Problem 2P

(a)

To determine

An example of a $3\times 3$ doubly stochastic matrix and check if the matrix is symmetric or not. Also, check whether all the $3\times 3$ doubly stochastic matrices are symmetric or not.

(b)

To determine

To prove: The product of the matrix $\left[\begin{array}{ccc}0.1& 0.3& 0.6\\ 0.6& 0.1& 0.3\\ 0.3& 0.6& 0.1\end{array}\right]$ and the matrix $\left[\begin{array}{ccc}0.2& 0.3& 0.5\\ 0.3& 0.5& 0.2\\ 0.5& 0.2& 0.3\end{array}\right]$ is doubly stochastic.

(c)

To determine

To prove: The matrix $\left[\begin{array}{c}\frac{1}{3}\\ \frac{1}{3}\\ \frac{1}{3}\end{array}\right]$ is a stable distribution for the matrix $\left[\begin{array}{ccc}0.2& 0.3& 0.5\\ 0.3& 0.5& 0.2\\ 0.5& 0.2& 0.3\end{array}\right]$.