I'm exploring the interaction between different types of stochastic matrices and came across a scenario that I find quite intriguing. Consider a nonnegative $(n \times n) $ matrix ($A$) that is row-stochastic (each row sums to 1) but not column-stochastic (at least one column does not sum to 1). My question is about the possibility of constructing a product matrix ($AB$) that is doubly stochastic, where ($B$) is any nonnegative stochastic matrix of the same dimension.
From an intuitive standpoint, it seems unlikely that the lack of column-stochastic property in ($A$) can be compensated for by any choice of ($B$), allowing ($AB$) to fulfill the criteria for being doubly stochastic. However, I'm uncertain if there might exist specific conditions or examples where this intuition does not hold.
Is there a formal proof or counterexample that can confirm or deny the possibility of ($AB$) being doubly stochastic under the given conditions?