 Solution

Reading positive entries as directed edges, the communicating class decomposition isFor the first class, paths $1\to 5\to 8\to 9\to 1$ establish mutual reachability, and every transition from one of these states stays in this set. States 2 and 3 communicate, but state 2 can enter the first class and never return. States 4,6,7 communicate through $4\to 6\to 7\to 4$, but state 6 can enter state 2 with no route back. These paths also exclude any larger communicating class. In a finite Markov chain, the closed class is recurrent and the two open classes are transient.