TITLE: Is there a way of determining if two matrices have any relations?
QUESTION [5 upvotes]: 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!

REPLY [3 votes]: 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.