1. **Problem Statement:** Verify whether the given Markov chain transition matrix $$P = \begin{pmatrix} 0.3 & 0.7 & 0 \\ 0.1 & 0.4 & 0.5 \\ 0 & 0.2 & 0.8 \end{pmatrix}$$ is irreducible. 2. **Definition:** A Markov chain is irreducible if it is possible to get from any state to any other state in a finite number of steps. 3. **Step 1: Check connectivity between states:** - From state 1 to state 2: direct transition with probability 0.7 (non-zero). - From state 2 to state 3: direct transition with probability 0.5 (non-zero). - From state 3 to state 2: direct transition with probability 0.2 (non-zero). 4. **Step 2: Check if all states communicate:** - From state 1 to state 3: can go 1 → 2 → 3 (since 1 to 2 and 2 to 3 are possible). - From state 3 to state 1: check if possible in some steps. - From 3 to 2 (0.2), and from 2 to 1 (0.1), so 3 → 2 → 1 is possible. 5. **Conclusion:** Since every state can be reached from every other state in a finite number of steps, the matrix is irreducible. **Final answer:** The given Markov chain transition matrix is irreducible.