Unique Stationary Distribution for Reducible Markov Chaine

Is it possible for a reducible markov chain to have a unique stationary distribution. Consider e.g. the markov chain with transition matrix below

$$A= \begin{pmatrix} 1 & 0 & 0 \\\ 0.2 & 0.7 & 0.1 \\\ 0.3 & 0.3 & 0.4 \end{pmatrix}$$ I believe A is a reducible markov chain ({1} and {2,3} are two distinct classes of states). Once we visit state 1 we are stuck with state 1. However if we try to calculate its unique distribution it comes as pi_s = [1 0 0]. And if we start with any arbitrary probability distribution, after a while it seems we will converge to pi_s. I am not sure am I missing something?

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