Let $S_i$ be a Markov chain with transition Matrix $ P $ $ (0 ≤ p ≤ 1) $
\begin{equation*} P = \begin{pmatrix} 1-2p & 2 p& 0 \\ p & 1-2p & p \\ 0 & 2p & 1-2p \end{pmatrix} \end{equation*}
-For which values of p is $S_i$ ergodic?
-What is the stationary distribution?
I know that a Markov chain is called ergodic if it is possible to go from one state to every state (not necessarily in one move). So for $p \neq 0$ $S_i$ is ergodic.
When I try to find the stationary distribution then i get $(0,0,0)$ which doesn't seems right because summing these values must give us $1$.
Any help would be much appreciated..