Classify the states of this Markov chain which is given by a transition matrix

Classify the states of the Markov chain which is given by the transition matrix

$$M= \begin{pmatrix} 0 & 1/2 & 1/2 & 0 \\ 1/3 & 0 & 0 & 2/3\\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix}$$

It's important for me to fully understand how this is done because I'm pretty sure this will be asked in a test soon :s

I hope you can tell me if I did it correctly and if not how to do it correctly?

I start by converting this matrix to a graph because it's easier for me to classify the states:

Accessible states: $(a \rightarrow b),(a \rightarrow c),(b \rightarrow a),(b \rightarrow d),(c \rightarrow a),(d \rightarrow c)$

Communicative states: $(a \leftrightarrow b),(a \leftrightarrow c)$ ,(and thus also $(b \leftrightarrow c)$)

Thus we have $2$ equivalence classes: $C_1=\left\{a,b,c\right\}$ and $C_2=\left\{d\right\}$

Since we have $2$ equivalence classes, the Markov Chain is not irreducible.

Since states $a,b,c$ are in the same equivalence class, e.g. communicate with each other, we can conclude that they are recurrent states.

But state $d$ is also recurrent because it's impossible to not reach this state again from another state.

So there is no state which is transient and there is no state which is absorbable.

We say that state $j$ is accessible from state $i$ , written as $i \to j$, if $p^{(n)}_{ij}>0$ for some $n$. Note that $n$ need not be $1$, $i$ can reach $j$ in a few steps.
Note that we can reach $d$ from $b$.
No state is absorbing state since no state has an arrow that loop back to itself with probability $1$.