For any $A, B \in SL(2, F)$, does knowing $\operatorname{tr}A$, $\operatorname{tr}B$, and $\operatorname{tr}AB$ specify $A$ and $B$?

Question

In title, $F$ denotes a field. Does knowing the trace of two matrices and their product specify those two matrices? Up to some equivalence, perhaps?

Answer 1

You could replace $A$ and $B$ with conjugates $CAC^{-1}$ and $CBC^{-1}$, with the effect of replacing $AB$ with $CABC^{-1}$, so none of the traces change. So at best you could specify $A,B$ only up to conjugacy in this manner.

Answer 2

No. For instance, when $A=I$ and $\operatorname{tr} B=2$, the matrix $B$ can have two possible Jordan forms, namely, $I$ and $\pmatrix{1&1\\ 0&1}$. There isn't enough information to pin down the true one.