# Invertible matrices problem

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?

## Answers and Replies

Staff Emeritus
Homework Helper
Almost. The last step isn't quite correct.

Homework Helper
Gold Member
Verrrry close. Check your last step.

Precursor
So it just ends here?

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

Staff Emeritus
Homework Helper
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:

$$-I$$ ?