Chapter 9.2, Problem 48E

Given the transition matrix

$P=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\begin{array}{c}1\\ 0\\ 0\end{array}\begin{array}{c}0\\ 1\\ 0\end{array}\right]$

(A) Discuss the behavior of the state matrices $S_1,S_2,S_3,\mathrm{....}$ for the initial-state matrix $S_0=\left[.2\text{.3 }\text{.5}\right].$

(b) Repeat part (A) for $S_0=\left[\frac{1}{3}\frac{1}{3}\frac{1}{3}\right].$

(C) Discuss the behavior of $P^k,k=2,3,4,\mathrm{....}$.

(D) Which of the conclusions of Theorem 1 are not valid for this matrix? Why is this not a contradiction?