Chapter 9.3, Problem 59E

An absorbing Markov chain has the following matrix $P$ as a standard form:

$\begin{array}{l}\begin{array}{cccc}A& B& C& D\end{array}\\ P=\begin{array}{c}A\\ B\\ C\\ D\end{array}\left[\begin{array}{cccc}1& 0& 0& 0\\ .2& .3& .1& .4\\ 0& .5& .3& .2\\ 0& .1& .6& .3\end{array}\right]\left[\begin{array}{cc}I& 0\\ R& Q\end{array}\right]\end{array}$

Let $w_k$ denote the maximum entry in $Q^k$. Note that $w_1=.6$.

(A) Find $w_2,w_4,w_8,w_{16}$, and $w_{32}$ to three decimal places.

(B) Describe $Q^k$ when $k$ is large.