Chapter 8, Problem 1P

(a)

To determine

An example of a $2\times 2$ doubly stochastic matrix and check if the matrix is symmetric or not.

(b)

To determine

To prove: Every $2\times 2$ doubly stochastic matrix is symmetric.

(c)

To determine

To prove: The product of the matrix $\left[\begin{array}{cc}0.3& 0.7\\ 0.7& 0.3\end{array}\right]$ and the matrix $\left[\begin{array}{cc}0.4& 0.6\\ 0.6& 0.4\end{array}\right]$ is doubly stochastic.

(d)

To determine

To prove: The matrix $\left[\begin{array}{c}\frac{1}{2}\\ \frac{1}{2}\end{array}\right]$ is a stable distribution for the matrix $\left[\begin{array}{cc}0.3& 0.7\\ 0.7& 0.3\end{array}\right]$.