Solution
= Solution
\b[False.] On the two-state space, take
$$
P=
\begin{pmatrix}1&0\\0&1\end{pmatrix},
\qquad
Q=
\begin{pmatrix}\tfrac12&\tfrac12\\0&1\end{pmatrix}.
$$
Every positive entry of $P$ remains positive in $Q$. Under $P$, both singleton states are closed <communicating classes>, so neither state is a <transient state>. Under $Q$, however, state one eventually moves to the absorbing state two and can never return after doing so. State one is therefore transient.