Note_1

Given an event $\beta$ it is clear we can say MATH since $P_{ik}$ is the probability of going from state $i$ to state $k$ in one step. This works since we assume the Markov property which says the probability of transition to next state depends only on current state and not on any earlier state (for an order 1 Markov chain).

Note_2

We know that we can find such a row in $B$ using the following argument: Assume that there is no row in $B$ which sums to less than $1$. This means $B$ is an irreducable transitional probability matrix. However this is a submatrix of an original probability transition matrix which is irreducable, meaning it has no closed subsets. Hence $B$ can not be irreduacble (closed). Therefore, we can find at least one row in $B$ which sums to less than 1. (Matrix $B$ is similar to a $Q$ matrix, it has at least one row which sums to less than $1$).