Is there a way of determining if two matrices have any relations?

Question

If I have two matrices A and B, is there a way to determine if ANY relations exist?

e.g. $AAB^{-1}=B^{-1}ABBA^{-1}, BABAB^{-1}B^{-1}A^{-1}=ABAAB$, etc

Basically that every permutation of A, B and their inverses yields a unique matrix (apart from cases where a matrix is adjacent to it's inverse, which of course simplify)

Thanks!

Answer 1

For general matrices $A$ and $B$? No chance whatsoever.

Even the case $A,B\in SL(2,\mathbb{R})$ is nontrivial; see for instance this paper for a few examples of sufficient conditions.

Answer 2

When the matrices are invertible you may narrow down the search if they have different values of their determinant since $\det(A)^{n_A} \det(B)^{n_B}=1$, with $n_A$ and $n_B$ denoting the total number of appearances of $A$ and $B$, respectively, and with sign.