I need to show that the markov chain that has transition matrix written below is irreducible. \begin{bmatrix} 0.2 & 0.5 & 0.1 & 0.1 & 0.1 \\ 0.2 & 0.5 & 0.3 & 0 & 0 \\ 0.2 & 0 & 0.4 & 0.4 & 0 \\ 0.2 & 0 & 0.2 & 0.4 & 0.2 \\ 0.2 & 0 & 0 & 0.1 & 0.7 \end{bmatrix}
Is it enough for me to say for $n = 1,2,3,4,5$ we have that $\mathbb{P}(X_1 = n | X_0 =1) > 0$ and $\mathbb{P}(X_1 = 1 | X_0 =n) > 0$? Hence its irreducible. Is there any other (easier) way to show irreducibility?
Also, to calculate the stationary distribution, is the fastest/most convenient way to use $\pi P = \pi$?