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Let $A$ be a square right-stochastic matrix, so that $A$ has nonnegative entries and each row sums to unity.

For an invertible square matrix $B$, the product $A B$ is also right-stochastic. Must it be that $B$ itself is right-stochastic? I don't know, but the converse certainly holds since the product of right-stochastic matrices is right-stochastic.

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The answer is no. As an example, take $$ A = \frac 12 \pmatrix{1&1\\ 1&1}, \quad B = \pmatrix{1 & 1/2\\ 0&1/2}. $$ $B$ is not right-stochastic, but $AB = A$ is right-stochastic.

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  • $\begingroup$ Thank you; this is a very helpful answer. I forgot to write that $B$ is invertible. (The question is now updated.) Do you still believe there's a counter example? $\endgroup$
    – user551504
    Commented Feb 24, 2022 at 1:41
  • $\begingroup$ The answer is still no, see my edit $\endgroup$ Commented Feb 24, 2022 at 2:06
  • $\begingroup$ Much appreciated! $\endgroup$
    – user551504
    Commented Feb 24, 2022 at 2:07

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