[Math] How to compute the stationary distribution of a $2\times 2$ transition probability matrix more easily

Question

$P=\left(\matrix{0.7 \ \ \ \ 0.3\\0.6 \ \ \ \ 0.4}\right)$ be the transition probability matrix of finite markov chain wtih 2 states, $1,2$ and what is the probability that in long run, $P(X_n=1)$?

I know how to find stationary distribution but the solution suggest that it is equal to $\frac{0.6}{0.6+0.3}$ without using any stationary distribution and i don't quite get how it gets that.

Answer 1

You could try to show the following:

The stationary distribution of the Markov chain with transition matrix $\begin{pmatrix}1-a & a\\ b & 1-b\end{pmatrix}$ is proportional to $\begin{pmatrix}b\\ a\end{pmatrix}$, for every $a$ and $b$ in $(0,1)$.

Of course, such a characterization is specific to the two-states case.