Prove that the Markov chain is irreducible and determine the steady state probabilities. What is the average number of busy terminals?

Assume a computer system has three terminals. Each terminal is either busy or idle. The system can be in one of the following four states: all terminals are idle (state 0), one terminal is busy (state 1), two terminals are busy (state 2) and all terminals are busy (state 3). Observing its states at specific time of the day, the system behaves like a homogeneous discrete Markov chain with the transition probability matrix:

0.2 0.4 0.3 0.3 0 0.5 0.2 0.4 0 0.2 0.2 0.3 0.8 0 0.2