What justifies this equality?

Question

Suppose $P$ is a stochastic matrix with equal rows, with $P_{ij}=p_j$. By induction on $n$,

$$P_{ij}^n=\sum_k P_{ik}^{n-1}P_{kj}=\sum_k p_k p_j=p_j$$

What happens in the second equality? What justifies $P_{ik}^{n-1}$ being equal to $p_k$?

Answer 1

You can prove that if P if stochastic with equal rows, than $P^n=P$ for all $n \geq 2$