Limiting Markov transition matrix

Question

If I have a transition matrix $P$ for $S$ states and I know that I can write it as $$P= \begin{bmatrix} A & B \\ 0 & I_d \end{bmatrix} $$ where $I_d$ is the identity matrix for the absorbing states (there is at least one absorbing state), $A$ is the probability transition from transient states to transient states and $B$ is the probability from transient states to absorbing states. I know that the limiting matrix $P^*$ exists and has the form $$P^* = \begin{bmatrix} 0 & B^* \\ 0 & I_d \end{bmatrix}. $$ Is it true that $B^*$ is such that for all file $j$ of the matrix $\sum_{i \in S} p_{ji} =1 $? where $p_{ij}$ is the probability to being in the state $i$ given the actual state $j$.