Solution
= Solution
{c}
The chain alternates between the classes $\{1,2\}$ and $\{3,4\}$. It is <irreducible> and has period two, since no odd return is possible and each state has a positive two-step return <probability>. Matrix multiplication gives
$$
\boxed{P^2=\begin{pmatrix}5/9&4/9&0&0\\4/9&5/9&0&0\\0&0&5/9&4/9\\0&0&4/9&5/9\end{pmatrix}.}
$$
The even-time sampled chain stays in its initial class and moves between its two states with <probability> $4/9$ at each sampled step.