Chapter 9.1, Problem 74E

In Problems 71-74, given the transition matrix $P$ and initial-state matrix $S_0$, find $P^4$ and use $P^4$ to find $S_4$.

$\begin{array}{l}\begin{array}{ccc}A& B& C\end{array}\\ P=\begin{array}{c}A\\ B\\ C\end{array}\left[\begin{array}{ccc}0& 1& 0\\ .8& 0& .2\\ 1& 0& 0\end{array}\right];S_0=\left[\begin{array}{ccc}.4& .2& .4\end{array}\right]\end{array}$