Exercise 6

Suppose that $A$ is a stochastic matrix and that $\xvec$ is a probability vector. We would like to explain why the product $A\xvec$ is a probability vector.

1. Explain why $\xvec=\threevec{0.4}{0.5}{0.1}$ is a probability vector and then find the product $S\xvec$.

2. More generally, if $\xvec$ is any probability vector, what is the product $S\xvec$?

3. If $A$ is a stochastic matrix, explain why $SA=S$.

4. Explain why $A\xvec$ is a probability vector by considering the product $SA\xvec$.