I am faced with the following question: given a Markov Chain with three possible states, $1,2,3$, and transition rates:
$$\alpha(1,2)=\alpha(2,3)=1,\:\alpha(2,1)=\alpha(3,2)=4$$
and the assumption that $X_0=1$, what is the probability that $X_t=3$ for all $t\in[2,3]$?
I have already found the probability that $X_2=3$ by deriving the transition matrix $P_t$ (which I got by calculating eigenvalues and solving for $P_t=e^{tA}=Qe^{tD}Q^{-1}$).
$$P_t=\left[\begin{array}{rrr} \frac{16}{21}&\frac{4}{21}&\frac{1}{21}\\ \frac{16}{21}&\frac{4}{21}&\frac{1}{21}\\ \frac{16}{21}&\frac{4}{21}&\frac{1}{21}\\ \end{array}\right] +e^{-3t}\left[\begin{array}{rrr} \frac{1}{6}&-\frac{1}{12}&-\frac{1}{12}\\ \frac{1}{3}&\frac{1}{6}&\frac{1}{6}\\ \frac{1}{6}&-\frac{1}{12}&-\frac{1}{12}\\ \end{array}\right] +e^{-7t}\left[\begin{array}{rrr} \frac{1}{14}&-\frac{3}{28}&\frac{1}{28}\\ \frac{3}{7}&\frac{9}{14}&\frac{4}{14}\\ \frac{4}{7}&-\frac{6}{7}&\frac{2}{7}\\ \end{array}\right] $$
I'm stuck as to where to go from here. I figure I need to somehow integrate over $[2,3]$, but I'm not sure precisely how to do that. Any suggestions would be sincerely appreciated. Thanks.