Let [ A= \left [{\begin{array}{cc} 3 & 1 \\ 0 & 1 \\ \end{array}} \right ] ] and [ B=\left [{\begin{array}{cc} 1 & 0 \\ 1 & 2 \\ \end{array}} \right ] ]

I want to show that the only elements of the semigroup generated by $A$ and $B$ that have integer eigenvalues are elements of the form $A^n$ and $B^n$, $n \in \mathbb{N}$. I have tried every way that I can think of. Is it possible that a problem like this is undecidable?