# Invertible matrices problem

1. Feb 28, 2010

### Precursor

1. The problem statement, all variables and given/known data
For the invertible matrices A, B and A-B, simplify the expression $$(A - B)^{-1}A(A^{-1} - B^{-1})B$$.

2. Relevant equations
properties of invertible matrices

3. 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?

2. Feb 28, 2010

### vela

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

3. Feb 28, 2010

### LCKurtz

Verrrry close. Check your last step.

4. Feb 28, 2010

### Precursor

So it just ends here?

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

5. Feb 28, 2010

### vela

Staff Emeritus
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.

6. Feb 28, 2010

### Precursor

Ok, so then the answer is:

$$-I$$ ?

7. Feb 28, 2010

### LCKurtz

8. Feb 28, 2010

Thanks