Exercise6

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 $$S\xvec\text{.}$$

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

3. If $$A$$ is a stochastic matrix, explain why $$SA=S\text{.}$$

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

in-context