Solving AxB = (B-1A-1)-1: Inverse Matrix Proof

[Cannonstar]

I'm having a bit of a struggle with my assignment.

I'm supposed to find what is x in AxB = (B^-1A^-1)^-1 .

I'm stumped at what to do with this. My friend said that x is I (identity matrix), but he is unable to prove it as well. My linear algebra class just recently started doing this topic and I haven't fully absorbed the subject yet.

Any hints or tips would be helpful though.

Thanks!

 

[shmoe]

Hi, you should try to simplify the right hand side, starting with the outermost -1. What rules do you have for the inverse of a product of matrices?

 

[HallsofIvy]

(B^-1A^-1)^-1 is the matrix C such that C(B^-1A^-1)= I. Since you have that equal to AxB, how do you get AxB(B^-1A^-1)= I??