Markov chain explanation

Question

I'm reading Norris' book on Markov chains and am unsure about how the relation $P^{n+1} = P^n P$ was used to obtain the last equation in the following example.

Example 1.1.4

Sorry it's probably an elementary question but I'm new to Markov chains so if someone could explain it I'd really appreciate it, thanks

Answer 1

$p_{11}^{(n+1)}$ is the $(1,1)$ entry of the matrix $P^{n+1}$. If you think about how matrix multiplication works, you can see that this is equal to the dot product of the first row of $P^n$ (which has entries $p^{(n)}_{11}$ and $p^{(n)}_{12}$) with the first column of $P$ (which has entries $1-\alpha$ and $\beta$).