Consider a Markov chain on $\Omega=\{1,2,3,4,5\}$ specified by the following transition matrix.
$$ \left[ \begin{array}{ccccc} 0.2& 0.2 &0.5& 0&0.1\\ 0.3 &0& 0.3& 0.4&0\\ 0.2 &0.3& 0.2 &0.3&0\\ 0& 0& 0& 0.8&0.2\\ 0&0&0&0&1\\ \end{array} \right] $$
Am I correct that:
(1) there are three communicating classes: $\{1,2,3\}$, $\{4\}$, $\{5\}$.
(2) State 5 is an absorbing state
(3) $\{4\}$ is transient,
(4) $\{1,2,3\}$ and $\{5\}$ are recurrent.