##### 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.

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

More generally, if \(\xvec\) is any probability vector, what is the product \(S\xvec\text{?}\)

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

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