Pre-Algebra · Superchallenge problem

Two different digits $A$ and $B$, each from $1$ to $9$, are used to form the two-digit numbers $\overline{AB}$ and $\overline{BA}$ (so $\overline{AB}=10A+B$ and $\overline{BA}=10B+A$). Suppose $\overline{AB}+\overline{BA}=121$. Find the sum of all possible values of the product $A\times B$.