Is a Markovian chain irreducible when one state does not have a recursive path?

Question

Let be the following homogeneous Markovian chain with three state:

\begin{pmatrix} 1/2 & 1/4 & 1/4 \\ 2/3 & 0 & 1/3\\ 3/5 & 1/5 & 1/5 \end{pmatrix}

Is this Markovian chain irreducible?

From wikipedia:

A Markov chain is said to be irreducible if its state space is a single communicating class; in other words, if it is possible to get to any state from any state.

But here, there is no recursive path from 2 to 2, my answer would be no, but is it true?

Answer 1

It is irreducible by the wiki definition. You can go 2 -> 1 -> 2.