Chapter 9.3, Problem 53E

The following matrix $P$ is a nonstandard transition matrix for an absorbing Markov chain:

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

To find a limiting matrix for $P$, follow the steps outlined below.

Step 1 Using a transition diagram, rearrange the columns and rows of $P$ to produce a standard form for this chain.

Step 2 Find the limiting matrix for this standard form.

Step 3 Using a transition diagram, reverse the process used in Step 1 to produce a limiting matrix for the original matrix $P$.