Consider a Markov chain {Xn; n ∈ N} taking values in the finite state E = {1, 2, 3, 4, 5, 6}, with a transition matrix P whose off-diagonal entries are given by
1. Find the diagonal entries of P.
2. Show that E can be partitioned into three equivalence classes to be specified, of which one (T ) is transient and two (R1 and R2) are recurrent.
3. Find an invariant probability whose support is R1 and another whose support is R2. Find all invariant probabilities.