Prove if there exists invertible 2x2 matrices A and B such that
A*B*A^-1*B^-1=2I₂
If you are asking if there exist invertible matrices A and B such that $\displaystyle ABA^{-1}B^{-1} = 2I$, then we can easily say there does not. Use determinants.
However I think you dont know determinants and thats why they have given a 2x2 exercise. So it is the same as saying, AB = 2BA, Now write down the matrices 4 variables and equate the matrices term by term. This will show you that A or B cant be invertible.