Is there a way to compute one matrix element of the exponential of a tridiagonal matrix without having to compute the rest of the elements?

Motivation: I'm trying to find the first passage time distribution from a master equation. I can impose an absorbing boundary at the threshold $n$, and the master equation with the new boundary condition is of the form $\frac{dp}{dt}=A p$ for a tridiagonal $n\times n$ matrix $A$. Then the first passage time distribution can be written as a particular matrix element of $\exp(A\,t)$. It takes forever for Mathematica to compute $\exp(A\,t)$ for large $n$, so I was wondering if there is a way to compute only the desired matrix element and not the whole matrix.