$$P^\ast= \begin{bmatrix} 0 & 0 & 1 & 0 & 0 & 0 & 0& 0 \\ 0 & 0 & 0 & 0 & \frac{1}{3} & 0 & 0 & \frac{2}{3} \\ 0& 0 & 0 & 0 & 1 & 0 & 0 & 0\\ \frac{2}{3} & \frac{1}{6} & 0 & 0 & 0 & \frac{1}{6} & 0 & 0 \\ \frac{2}{3} & 0 & \frac{1}{3} & 0& 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ \end{bmatrix}$$ Determine all open and closed classes.
I am struggling to get my head around the classification of states in a Markov Chain $\{X_t\}_{t\geq0}$. I know that $\{6,7\}$ is a closed class as the chain goes to state 7 when initial state is 6 and vice versa. But what about the rest? And would 8 be included in this $\{6,7\}$ class? I am so confused, any help is appreciated!
Update: I have figured out that $C_1=\{6,7,8\}$ is a closed class , and so is $C_2=\{1,3,5\}$ . And a guess would be $C_3=\{4\}$ is open and $C_4=\{2\}$ ?
Closed classes are $\{1,3,5\}$ and $\{6,7\}$. Open classes are $\{2\}$, $\{4\}$ and $\{8\}$.