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We know that a matrix is irreducible if it is not similar via a permutation to a block upper triangular matrix.

Is the transition matrix of a irreducible Markov chain irreducible?

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    $\begingroup$ Yes. You can do a proof by contradiction by showing that a block upper triangular matrix cannot be the transition matrix of an irreducible Markov chain; as a further hint (I assume, with little justification, you are looking for hints rather than a straight answer), can you get back to state 1 from any higher-indexed state if you have a block upper triangular transition matrix? $\endgroup$ – jbowman Sep 14 '18 at 21:43

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