Chapter 8, Problem 3P

(a)

To determine

An example of a $4\times 4$ doubly stochastic matrix and check if the matrix is symmetric or not. Also, check whether all the $4\times 4$ doubly stochastic matrices are symmetric or not.

(b)

To determine

To prove: The product of the matrix $\left[\begin{array}{cccc}0.1& 0.2& 0.3& 0.4\\ 0.3& 0.4& 0.1& 0.2\\ 0.2& 0.3& 0.4& 0.1\\ 0.4& 0.1& 0.2& 0.3\end{array}\right]$ and the matrix $\left[\begin{array}{cccc}0.2& 0.3& 0.1& 0.4\\ 0.3& 0.2& 0.4& 0.1\\ 0.1& 0.4& 0.2& 0.3\\ 0.4& 0.1& 0.3& 0.2\end{array}\right]$ is doubly stochastic.

(c)

To determine

To prove: The matrix $\left[\begin{array}{c}\frac{1}{4}\\ \frac{1}{4}\\ \frac{1}{4}\\ \frac{1}{4}\end{array}\right]$ is a stable distribution for the matrix $\left[\begin{array}{cccc}0.2& 0.3& 0.1& 0.4\\ 0.3& 0.2& 0.4& 0.1\\ 0.1& 0.4& 0.2& 0.3\\ 0.4& 0.1& 0.3& 0.2\end{array}\right]$.