Specify the classes of the Markov chain, and state whether they are recurrent or transient.
\begin{align*} P_4= \begin{bmatrix} \frac{1}{4} & \frac{3}{4} & 0 & 0 & 0\\ \frac{1}{2} & \frac{1}{2} & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & \frac{1}{3} & \frac{2}{3} & 0\\ 1 & 0 & 0 & 0 & 0\\ \end{bmatrix}. \end{align*}
I have done three problems like this but I'm not sure if I'm correct. I'm just making sure I understand the idea of recurrent and transient classes as opposed to states. For this problem, I have
$\{0, 1\}, \{2\}, \{3\}, \{4\}$ for the classes. The first two are recurrent and the last two are transient.
Is this correct? If not, can someone explain what's wrong?
I obtained my solutions by drawing a diagram.