Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 2 / 18H / a / Solution

True. If the Markov chain with transition matrix $P$ is irreducible, then for every pair of states $i,j$ there is a pathwith $P_{i_{r-1}{i}_r}>0$ at every step. The support assumption gives $Q_{i_{r-1}{i}_r}>0$ for every edge of the same path. Thus every state can reach every other state under $Q$, so the second chain is also an irreducible Markov chain.