TITLE: Proving or disproving product of two stochastic matrices is stochastic
QUESTION [5 upvotes]: Let $P$ and $Q$ be two stochastic matrices. Does the product $PQ$ have to be stochastic? Prove or disprove.

What Im thinking is that since matrix multiplication is only defined for two matrices $A$ and $B$ where $A$ has the same amount of columns as $B$ has rows and vice versa. Therefore the product of any two stochastic matrices $P$ and $Q$ can not be stochastic. This is not much of a proof though.

REPLY [3 votes]: This answer is a bit late, but I did want to point t out that you can easily get the result for row-stochastic matrices from column-stochastic matrices (and vice versa).
Let A and B be two row-stochastic matrices and suppose we know the product of column stochastic matrices is column-stochastic. Observe that,
$$AB = ((AB)^T)^T = (B^TA^T)^T$$
by properties of transpose of a matrix. Let us consider $B^TA^T$. It is easy to see that the transpose of a row-stochastic matrix is column-stochastic by definition (and vice versa). Thus, $B^T$ and $A^T$ are column stochastic and by our assumption, it must then be the case that $B^TA^T$ is column-stochastic. Since $B^TA^T$ is column-stochastic, then it's transpose $(B^TA^T)^T = AB$ is row stochastic.