Classifying the state space of the Markov chain

Question

Classify the state space of the Markov chain with following transition probability matrix into communicating classes:

$$ \left[ \begin{array}{cccc} 0.2& 0.2 &0.6& 0\\ 0.6 &0.2& 0.1& 0.1\\ 0.3 &0.3& 0.3 &0.1\\ 1.0& 0& 0& 0 \end{array} \right] $$

My Solution:

Doesn't this Markov chain have only one communicating class $\{0,1,2,3\}$ since $0\rightarrow1\rightarrow2\rightarrow3$, hence this is an irreducible Markov chain?

Answer 1

Yes, your answer is right. This Markov chain has got only 1 communication class and this is irreducible Markov chain