Chapter 9, Problem 28RE

Show that $S=\left[\begin{array}{cccc}x& y& z& 0\end{array}\right]$, where $0\le x\le 1$, $0\le y\le 1$, $0\le z\le 1$, and $x+y+z=1$, is a stationary matrix for the transition matrix

$\begin{array}{l}\begin{array}{cccc}A& B& C& D\end{array}\\ P=\begin{array}{c}A\\ B\\ C\\ D\end{array}\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ .1& .3& .4& .2\end{array}\right]\end{array}$

Discuss the generalization of this result to any absorbing chain with three absorbing states and one nonabsorbing state.