Chapter 9.2, Problem 53E

The transition matric for a Markov chain is

$P=\left[\begin{array}{ccc}.1& .5& .4\\ .3& .2& .5\\ .7& .1& .2\end{array}\right]$

Let $M_k$ denote the maximum entry in the second column of $P^k$. Note that $M_1=.5$.

(A) Find $M_2,M_3,M_4$, and $M_5$ to three decimal places.

(B) Explain why $M_k\ge M_{k+1}$ for all positive integers $k$.