# Homework Help: Inverse of the difference of two matrices A and B

1. Oct 4, 2008

### yukawa

1. The problem statement, all variables and given/known data
Show that
1/(A-B) = (1/A) + (1/A)B(1/A) + (1/A)B(1/A)B(1/A) + (1/A)B(1/A)B(1/A)B(1/A)+.....

where A and B are matrices whose inverse exist.

3. The attempt at a solution
I tried to start from the LHS by pulling out (1/A):
LHS = (1/A)[1 + B(1/A) + B(1/A)B(1/A) + B(1/A)B(1/A)B(1/A)+.....]
= (1/A)*{1/[1-B(1/A)]} where I have used equation of GP sum to infinity
= 1/(A - AB(1/A))
However, i can't get the expression in RHS since A and B are not commute.

Are there any other possible approach to this problem?

Any help would be great~thanks~

Last edited: Oct 5, 2008
2. Oct 5, 2008

### HallsofIvy

Re: Matrix

So, more simply, (1/A)(1/1-(B/A))= (1/A)[(B/A)+ (B/A)2+ (B/A)3+ ...)

Do you know the proof of the sum of a geometric series?>

3. Oct 5, 2008

Re: Matrix

The geometric series approach came to mind as well, but since these are matrices some care needs to be taken.
If this is simply a formal problem the suggestion makes sense: usually, however, it is not enough to go through the mechanics - some conditions on the norms of the matrices are required.
This seems to be a rather poorly worded problem
(BUT, I agree with the suggestion of HallsofIvy - that seems to be the intent here)

4. Oct 5, 2008

### HallsofIvy

Re: Matrix

You cannot assume that the formula for the sum of a geometric series of real numbers, but you can copy the proof- being careful that you don't assume commutativity.