[Math] Are these transient or recurrent states in a Markov chain

Question

I have the following transition matrix for a Markov chain with states $A, B, C, D, E$:

$$
\left| \begin{array}{ccc}
0 & 0 & \frac{1}{2} & \frac{1}{2} & 0 \\
\frac{1}{2} & 0 & \frac{1}{2} & 0 & 0 \\
\frac{1}{2} & 0 & 0 & \frac{1}{2} & 0 \\
0 & \frac{1}{2} & 0 & 0 & \frac{1}{2} \\
0 & 0 & 0 & 0 & 1 \end{array} \right|
$$

State $E$ is an absorbing state, but I am wondering how to classify the other states. After drawing out the transition diagram, it seems that all the other states are transient states as eventually we will end up in state $E$. There are no states where we 'get stuck' alternating back and forward. Is this correct?

Answer 1

I seem to recall a theorem, in which if an absorbing state is accessible from a communicating class $C$, then all states in $C$ are transient.