Solving Backwards Chapman-Kolmogorov Questions The question is
A flea hops on the vertices A, B and C of a triangle. Each hop takes it from one vertex
to the next and the times between successive hops are independent random variables,
each with an exponential distribution with mean $\frac{1}{\lambda}$. Each hop is equally likely to be
in the clockwise direction or in the anticlockwise direction. Find the probability that
the flea is at vertex A at a given time t > 0, starting from A at time t = 0.  
I have to solve this using Backwards Kolmogorov equations but it's just so messy that I don't have an intuition on what equations to find since there are $9$ of them...
I want $P_{AA}(t)$ and currently I have
$$P'_{AA}(t) = \lambda(P_{BA}(t) + P_{CA}(t) - 2P_{AA}(t))\\
P'_{BA}(t) = \lambda(P_{AA}(t) + P_{CA}(t) - 2P_{BA}(t))\\
P'_{CA}(t) = \lambda(P_{AA}(t) + P_{BA}(t) - 2P_{CA}(t)).
$$
 A: The ODE system observes the following equivalent matrix form
$$
\frac{\rm d}{{\rm d}t}\left(
\begin{array}{c}
P_{AA}\\
P_{BA}\\
P_{CA}
\end{array}
\right)=-\lambda\left(
\begin{array}{ccc}
2&-1&-1\\
-1&2&-1\\
-1&-1&2
\end{array}
\right)\left(
\begin{array}{c}
P_{AA}\\
P_{BA}\\
P_{CA}
\end{array}
\right),
$$
or
$$
\frac{{\rm d}\mathbf{p}}{{\rm d}t}=-\lambda Q\mathbf{p}
$$
for short. Here the symmetric matrix $Q$ yields the following orthogonal diagonalization
$$
Q=U\Lambda U^{\top},
$$
where
$$
U=\left(
\begin{array}{ccc}
-\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}\\
0&\frac{\sqrt{2}}{\sqrt{3}}&\frac{1}{\sqrt{3}}\\
\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}
\end{array}
\right)\quad\text{and}\quad\Lambda=\left(
\begin{array}{ccc}
3&&\\
&3&\\
&&0
\end{array}
\right).
$$
Therefore, if
$$
\mathbf{p}(0)=\mathbf{p}_0,
$$
the solution reads
$$
\mathbf{p}(t)=e^{-\lambda Qt}\mathbf{p}_0=Ue^{-\lambda\Lambda t}U^{\top}\mathbf{p}_0=U\left(
\begin{array}{ccc}
e^{-3\lambda t}&&\\
&e^{-3\lambda t}&\\
&&1
\end{array}
\right)U^{\top}\mathbf{p}_0.
$$
