In an irreducible Markov Chain all states belong to a single communicating class.

The given transition probability matrix corresponds to an irreducible Markov Chain. This can be easily observed by drawing a state transition diagram.

Alternatively, by computing $P^{(4)}$, we can observe that the given TPM is regular. This concludes that the given Markov Chain is irreducible. $$ P^{(4)} = \begin{matrix} 0.1576938 & 0.2583928 & 0.08312500 & 0.2327933 & 0.2588625\cr 0.1655115 & 0.2854474 & 0.11632500 & 0.2161569 & 0.1923158\cr 0.1500375 & 0.1895678 & 0.09953125 & 0.2075683 & 0.3465500\cr 0.1218750 & 0.2135125 & 0.10625000 & 0.2215688 & 0.3334688\cr 0.1277500 & 0.0615000 & 0.07750000 & 0.1892750 & 0.5437250\cr \end{matrix} $$

In an irreducible Markov Chain all states belong to a single communicating class.

The given transition probability matrix corresponds to an irreducible Markov Chain. This can be easily observed by drawing a state transition diagram.

Alternatively, by computing $P^{(4)}$, we can observe that the given TPM is regular. This concludes that the given Markov Chain is irreducible. $$ P^{(4)} = \begin{matrix} 0.1576938 & 0.2583928 & 0.08312500 & 0.2327933 & 0.2588625\cr 0.1655115 & 0.2854474 & 0.11632500 & 0.2161569 & 0.1923158\cr 0.1500375 & 0.1895678 & 0.09953125 & 0.2075683 & 0.3465500\cr 0.1218750 & 0.2135125 & 0.10625000 & 0.2215688 & 0.3334688\cr 0.1277500 & 0.0615000 & 0.07750000 & 0.1892750 & 0.5437250\cr \end{matrix} $$