Solution
= Solution
\b[True.] If the <Markov chain> with <transition matrix> $P$ is irreducible, then for every pair of states $i,j$ there is a path
$$
i=i_0,i_1,\ldots,i_m=j
$$
with $P_{i_{r-1}i_r}>0$ 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>.