I am currently working on a problem, where I need to know for square integer matrices, $A$ and $B$, whether or not there exists square integer matrices, $X$ and $Y$, such that $X(A-I)Y=B-I$, where $I$ is the identity matrix.
Is there any way to do this in Maple or any other software that you know? It can be done by writing it out as a huge system of linear equations, but that approach is both inefficient and horrible to work with.
Using the idea of Maestro Will: Necessarily $\det(A-I)$ divides $\det(B-I)\not= 0$ or $\det(B-I)=0$.
We calculate the Smith normal forms $A-I=SDT,B-I=U\Delta V$. Thus $(U^{-1}XS)D(TYV^{-1})=\Delta$. It suffices to solve the equation $XDY=\Delta$. where $d_i$ divides $d_{i+1}$ and $\delta_i$ divides $\delta_{i+1}$.
An instance without any solutions: $D=diag(2,24),\Delta=diag(4,12)$. One of the equations is $6x_{1,2}y_{2,1}+2x_{2,2}y_{2,2}=1$, that has no solutions in the integers.