As i presumed $|A_{i,j}|,|B_{i,j}|<1$ this does not matter since for $t<1$ obviously $t^2<t$ holds. @fedja is right that this does not work for their example, though I wonder if this only happens for some null subset of $\mathbb{R}^{m \times n}$. I would still be interested in a good upper bound for the first sum.