Markov chain on state space $S$, $A \subset S$ , $\forall i \in A, \sum_{j \in A} P_{ij} = 1$. Show that the Markov chain is not irreducible.

Question

A Markov chain on state space $S$, and suppose there is a strict subset $A \subset S$ such that for all $i \in A, \sum_{j \in A} P_{ij} = 1$. Show that the Markov chain is not irreducible.

My solution -:

Assume it's irreducible, then we know that an irreducible Markov chain with a finite state space is always recurrent. Which means $\sum_{n=1}^{\infty} P_{ii}^{(n)} = \infty$, and this is not possible from the given condition. Thus contradiction

Is this the correct approach or am I missing something?

Answer 1

Considering that (by definition) a Markov chain is irreducible if it is possible to get from any state to any other. Yes, there are many other definitions, but in this case it is worth using this one.
Now it is just left to understand, that you can NOT get out of the set $A$, based on the given condition on $P_{ij}$ (can you see it?).
This appears to be enough to conclude that the chain is NOT irreducible, as $A$ is a proper subset of $S$. All is left to do, is to formalize this.