Consider a markov chain with transition probability matrix given by
\begin{bmatrix}0.5 &0.5 & 0 \\0&0.5&0.5\\0.33 &0.33&0.33 \end{bmatrix}
let $p^n_{ij}$ denote n-step transition probability from state i to state j then without calculating left eigenvector show that
$1.\lim_{n\to\infty}p^n_{11}= \frac{2}{9} $ and
$2.\lim_{n\to\infty}p^n_{13}= \frac{1}{3} $
i only know how to solve it using left eigen vector , but i am not able to think of quicker way to solve this .