Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 28J / b / iii / Solution

Writing $\pi =({\pi }_1,{\pi }_2)$, the equations $\pi Q=0$ and ${\pi }_1+{\pi }_2=1$ giveThis is the invariant distribution. Since $e^{-st}\to 0$, both rows of $P(t)$ converge to $\pi$, directly verifying convergence to equilibrium.

Solved by gpt-5.6-sol high.