Let ${\{X_t | t\ge 0\}}$ be a Markov Process on a state space $S={\{1,2,3\}}$ with a generator $$ G=\begin{bmatrix} -1 & 0 & 1 \\ 3 & -4 & 1 \\ 2 & 0 & -2 \end{bmatrix}$$ (a) Find a semigroup ${\{ P_t | t \ge 0\}}$ and find a stationary distribution
(b) Calculate $P(X_{3t}=1 | X_{5t}=3, X_{0}=2, X_{4t}=3, X_{t}=2)$
(a) $G=B\;A\;B^{-1}$. I've got a formula that $P_t=\sum_{n=0}^{\infty} \frac{t^n}{n!}A^n=B(\sum_{n=0}^{\infty} \frac{t^n}{n!}A^n)B^{-1}$, so I'm diagonalizing my matrix G, substitute to my formula and that's all.
According to stationary distribution we have that $\pi G=0$, so $(\pi_1,\pi_2,\pi_3)G=(0,0,0)$, and from that I've found that $(\pi_1,\pi_2,\pi_3)=(2/3,0,1/3)$, because $\sum \pi_i=1$.
(b)I suppose that I need some formulas to calculate this probability but I can't recollect any.
Please, correct me where I'm wrong and help me with a subpoint (b).