Chapter 3.7, Problem 6EQ

To determine

$\begin{array}{l}\text{Given that P = }\left[\begin{array}{ccc}\frac{1}{2}& \frac{1}{3}& \frac{1}{3}\\ 0& \frac{1}{3}& \frac{2}{3}\\ \frac{1}{2}& \frac{1}{3}& 0\end{array}\right]\text{ is a transition matrix for a Markov chain with two states}\text{.}\\ {\text{If x}}_0\text{ = }\left[\begin{array}{c}120\\ 180\\ 90\end{array}\right]\text{ is the initial state for the population}\text{.}\\ \text{What proportion of the state 1 population will be in state 1 after two steps?}\end{array}$