Chapter 9.2, Problem 49E

The transition matrix for a Markov chain is

$P=\left[\begin{array}{ccc}1& 0& 0\\ .2& .2& .6\\ 0& 0& 1\end{array}\right]$

(A) Show that $R=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]$ and $S=\left[\begin{array}{ccc}0& 0& 1\end{array}\right]$ are both stationary matrices for $P$. Explain why this does not contradict Theorem $1\text{A}$.

(B) Find another stationary matrix for $P$. [Hint: Consider $T=aR+\left(1-a\right)S,\text{ where }0<a<1$.]

(C) How many different stationary matrices does $P$ have?