Why we can calculate the limiting probability of $\begin{pmatrix}0.6&0.2&0.1&0.1\\0.6&0&0.3&0.1\\0&0.6&0&0.4\\0&0&0.6&0.4\end{pmatrix}$ in this way?
Solution: Taking $\begin{pmatrix}\pi_0&\pi_1&\pi_2&\pi_3\end{pmatrix}(I-P)=$$\begin{pmatrix}\pi_0&\pi_1&\pi_2&\pi_3\end{pmatrix}\begin{pmatrix}-0.4&0.2&0.1&0.1\\0.6&-1&0.3&0.1\\0&0.6&-1&0.4\\0&0&0.6&-0.6\end{pmatrix}=0$ and we get the limiting probability
I don't quite get why it can be calculated in this way as state 2 is obviously not aperiodic so can this method being used? If not, how should one calculate it?