= Solution
Reading positive entries as directed edges, the <communicating class> decomposition is
$$
\boxed{\{1,5,8,9\}\text{ closed};\qquad\{2,3\}\text{ open};\qquad\{4,6,7\}\text{ open}}.
$$
For the first class, paths $1\to5\to8\to9\to1$ 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 $4\to6\to7\to4$, 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.
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