How to Simplify an Expression with Invertible Matrices?

[Precursor]

Homework Statement

For the invertible matrices A, B and A-B, simplify the expression $$(A - B)^{-1}A(A^{-1} - B^{-1})B$$.

Homework Equations

properties of invertible matrices

The Attempt at a Solution

$$(A - B)^{-1}A(A^{-1} - B^{-1})B$$
= $$(A - B)^{-1}(AA^{-1}B - AB^{-1}B)$$
= $$(A - B)^{-1}(IB - AI)$$
= $$(A - B)^{-1}(B - A)$$
= $$I$$

Am I correct?

 

[vela]

Almost. The last step isn't quite correct.

 

[LCKurtz]

Verrrry close. Check your last step.

 

[Precursor]

So it just ends here?

$$(A - B)^{-1}(B - A)$$

 

[vela]

No, you can simplify it. Consider this. Let C=B-A. Then C^-1=(B-A)^-1, but in your expression, you have (A-B)^-1, which isn't the same matrix.

 

[Precursor]

No, you can simplify it. Consider this. Let C=B-A. Then C^-1=(B-A)^-1, but in your expression, you have (A-B)^-1, which isn't the same matrix.

Ok, so then the answer is:

$$-I$$ ?

 

[LCKurtz]

 

[Precursor]

Thanks