Exercise 7

Using the results of the previous exercise, we would like to explain why $A^2$ is a stochastic matrix if $A$ is stochastic.

1. Suppose that $A$ and $B$ are stochastic matrices. Explain why the product $AB$ is a stochastic matrix by considering the product $SAB$.

2. Explain why $A^2$ is a stochastic matrix.

3. How do the steady-state vectors of $A^2$ compare to the steady-state vectors of $A$?