Solution
= Solution
Conditioning on the initial state gives
$$
\boxed{P(X_n=i)=\sum_{j\in S}\lambda_j(P^n)_{ji}=(\lambda P^n)_i.}
$$
Thus the row-vector distribution at time $n$ is $\lambda P^n$.
= Solution
Conditioning on the initial state gives
$$
\boxed{P(X_n=i)=\sum_{j\in S}\lambda_j(P^n)_{ji}=(\lambda P^n)_i.}
$$
Thus the row-vector distribution at time $n$ is $\lambda P^n$.