The fundamental matrix of a discrete time markov chain with absorbing states dictates the expected amount of time spent in each state $j$, given that you started in state $i$. The equation is $$S = (I-Q)^{-1}$$ where $Q$ is the transitive-state-only matrix (take out all rows and columns of absorbing states).
I want the expected amount of time spent in each state given a starting state, but ALSO given the place where you eventually absorb. Specifically, in my Markov chain, the first and last states are absorbing, so the fundamental matrix $S$ tells me the expected times given absorption in either state. I want the expected time given eventual absorption in a specific state (the first one).
I thought about making the last state non-absorbing, but it seems kind of arbitrary. I also thought about re-weighting everything so that the last state doesn't even exist, but that's also kind of silly. Thanks in advance!