Past exam of the probability course of the University of Oxford / 2014 / 3 / a / i / Solution

Write the stationary distribution as a row vector $\pi =(a,b,c,d)$. For the first transition matrix, $\pi P=\pi$ gives $a=a/2+d$, $b=a/2+b/2$, $c=b/2$ and $d=c$. Hence $b=a$ and $c=d=a/2$. Normalization yieldsBy the stated common-stationary distribution property, this is also the answer for the second transition matrix.