Markov Chains, Assistance will be much appreciated! 1) In the Markov chain linked below, in the long run, what fraction of the time does the chain spend in state 3? 
https://i.stack.imgur.com/tntbj.png
2) Find the probabilities of states  1 ,  2 , and  4  in the stationary distribution of the Markov chain  's' linked below. The label to the left of an arrow gives the corresponding transition probability.
https://i.stack.imgur.com/KXqdQ.png
 A: 1) $P = \begin{pmatrix} 
0.5 & 0.5 & 0 & 0 & 0 & 0  \\
0.5 & 0 & 0.5 & 0 & 0 & 0 \\
0 & 0 & 0.5 & 0.3 & 0.2 & 0 \\
0 & 0 & 0.3 & 0 & 0 & 0.7 \\
0 & 0 & 0.2 & 0 & 0.8 & 0 \\
0 & 0 & 0 & 0.7 & 0 & 0.3
\end{pmatrix}$
So the long run fraction corresponds to the probability distribution $\pi$ such that: $P^T \pi = \pi $. So we need to find eigenvector of $P^T$ corresponding to eigenvalue = 1. Such eigenvector is $\pi = (0, 0, 0.25, 0.25, 0.25, 0.25)$ (One can check this by checking that $P^T \pi = \pi$). So in the long run 25 percent of time chain spends in state 3.
2)
$P = \begin{pmatrix} 
0.2 & 0.4 & 0.4 & 0 & 0 & 0 & 0 \\
0.2 & 0 & 0 & 0.4 & 0.4 & 0 & 0 \\
0.2 & 0 & 0 & 0 & 0 & 0.4 & 0.4 \\
0 & 0.2 & 0 & 0.8 & 0 & 0 & 0 \\
0 & 0.2 & 0 & 0 & 0.8 & 0 & 0 \\
0 & 0 & 0.2 & 0 & 0 & 0.8 & 0 \\
0 & 0 & 0.2 & 0 & 0 & 0 & 0.8
\end{pmatrix}$
For this porblem stationary distribution (eigenvector for eigenvalue = 1) is $\pi = (0.04761905, 0.0952381 , 0.0952381 , 0.19047619, 0.19047619,
       0.19047619, 0.19047619) \ $ (Computed numerically)
