Chapter 9.3, Problem 57E

Show that $S=\left[\begin{array}{ccc}x& 1-x& 0\end{array}\right],0\le x\le 1$, is a stationary matrix for the transition matrix

$\begin{array}{l}\begin{array}{ccc}A& B& C\end{array}\\ P=\begin{array}{c}A\\ B\\ C\end{array}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ .1& .5& .4\end{array}\right]\end{array}$

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