Classify the states of the markov chain which is given by these two transition matrices $M_1$ and $M_2:$
$$M_1=\begin{pmatrix} \frac{1}{3} & \frac{2}{3} & 0 & 0\\ \frac{1}{2} & \frac{1}{2} & 0 & 0\\ \frac{1}{4} & 0 & \frac{1}{4} & \frac{1}{2}\\ 0 & 0 & 0 & 1 \end{pmatrix} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; M_2=\begin{pmatrix} 0 & \frac{1}{2} & \frac{1}{2} & 0\\ \frac{1}{3} & 0 & 0 & \frac{2}{3}\\ 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 \end{pmatrix}$$
I think we need to check the properties: rechability, communication and equivalence relation. Is that it or we need to check further properties?
But I'm rather more interested to know how you do it on this example because I'm confused we have two matrices given here instead of one. Will I need to individually check those properties for each matrix? And if not both matrices satisfy this property, thus I can conclude the markov chain doesn't satisfy this property?
Say we were checking reachability:
A state $j$ is reachable from state $i$, if there exists a number $n$ such that $p_{ij}(n)>0$
From this definition we see that the markov chain satisfies property reachability because in both matrices there is no column whith only zeroes.
Is this example correct I did?