If I have a Markov chain on states {0,1,2,3,4,5}
$$ \mathbf{a} = \matrix{~ & 0 & 1 & 2 & 3 & 4 & 5 \\ 0 & 1/3 & 0 & 2/3 & 0 & 0 & 0 \\ 1 & 0 & 0.25 & 0 & 0.75 & 0 & 0 \\ 2 & 2/3 & 0 & 1/3 & 0 & 0 & 0 \\ 3 & 0 & 0.20 & 0 & 0.80 & 0 & 0 \\ 4 & 0.25 & 0.25 & 0 & 0 & 0.25 & 0.25 \\ 5 & 1/6 & 1/6 & 1/6 & 1/6 & 1/6 & 1/6 } $$
and
$$ \mathbf{b} = \matrix{~ & 0 & 1 & 2 & 3 & 4 & 5 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0.75 & 0.25 & 0 & 0 & 0 \\ 2 & 0 & 1/8 & 7/8 & 0 & 0 & 0 \\ 3 & 0.25 & 0.25 & 0 & 1/8 & 3/8 & 0 \\ 4 & 1/3 & 0 & 1/6 & 0.25 & 0.25 & 0 \\ 5 & 0 & 0 & 0 & 0 & 0 & 1 } $$
I am supposed to find all communicating classes and specify which classes are transient and which are recurrent.
I found the classes for a to be: {0,2},{1,3},{4,5}
and the classes for b to be: {0},{5},{1,2},{3,4}
but I am having a hard time showing which are recurrent and which are transient. Can anyone help with this? Thanks in advance!