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 \(SAB\text{.}\)

  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 \(A\text{?}\)