Markov chain- recurrent and transient?

Question

In this Markov chain, are all the states recurrent since you can return back to every state starting anywhere? or are there any transient states?

A Markov chain has transition probability matrix $$P=\begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1/3 & 2/3 \\ 1 & 0 & 0 & 0 \\ 0 & 1/2 & 1/2 & 0 \\ \end{pmatrix}$$

Answer 1

You are right.All states are recurrent.