Factors of a stochastic matrix

Question

If $A B = C$ where matrix $C$ is stochastic (all entries are positive and all rows add to $1$) then is it necessary that both $A$ and $B$ be stochastic?

Answer 1

No, it is not necessary.

It may be that $A$ and $B$ are both stochastic:

$$ A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, B = \begin{pmatrix} 0.5 & 0.5 \\ 0.5 & 0.5 \end{pmatrix} \implies C = AB = \begin{pmatrix} 0.5 & 0.5 \\ 0.5 & 0.5 \end{pmatrix} . $$

It may be that $A$ and $B$ are not stochastic:

$$ A = \begin{pmatrix} 0 & 2 \\ 2 & 0 \end{pmatrix}, B = \begin{pmatrix} 0 & 0.5 \\ 0.5 & 0 \end{pmatrix} \implies C = AB = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} . $$

Answer 2

$$A=\begin{pmatrix} 2 & 0 \\ 0 & 2\end{pmatrix}, B = \begin{pmatrix}\frac12 & 0\\0 & \frac12\end{pmatrix}.$$