# Question about invertible matrices

icesalmon

## Homework Statement

Show that if A, B and A + B are all invertible and the same size
then A(A-1 + B-1)B(A + B)-1 = I
And what does the result say about A-1 + B-1

## The Attempt at a Solution

I start off by trying to reduce the LHS as much as I can so I multiply both sides on the right by ( A + B )
to get A(A-1 + B-1)B = (A+B)
(A-1 + B-1)B = A-1(A+B)
A-1 + B-1 = A-1(A+B)B-1
A-1 + B-1 = (A-1A + A-1B)B-1
A-1 + B-1 = ( IB-1 + A-1I )
A-1 + B-1 = A-1 + B-1

I've gotten the same thing on both sides, and yet again I don't even know what I've done.
thanks PF

Homework Helper

## Homework Statement

Show that if A, B and A + B are all invertible and the same size
then A(A-1 + B-1)B(A + B)-1 = I
A direct calculation on the left gives (I+ AB-1)B(A+B)-1= (B+ A)(A+ B)-1. And, of course, matrix addition is commutative.

And what does the result say about A-1 + B-1
Well, it says it is the inverse of ....

## The Attempt at a Solution

I start off by trying to reduce the LHS as much as I can so I multiply both sides on the right by ( A + B )
to get A(A-1 + B-1)B = (A+B)
(A-1 + B-1)B = A-1(A+B)
A-1 + B-1 = A-1(A+B)B-1
A-1 + B-1 = (A-1A + A-1B)B-1
A-1 + B-1 = ( IB-1 + A-1I )
A-1 + B-1 = A-1 + B-1

I've gotten the same thing on both sides, and yet again I don't even know what I've done.
thanks PF

Mentor

## Homework Statement

Show that if A, B and A + B are all invertible and the same size
then A(A-1 + B-1)B(A + B)-1 = I
And what does the result say about A-1 + B-1

## The Attempt at a Solution

I start off by trying to reduce the LHS as much as I can so I multiply both sides on the right by ( A + B )
If you do this, you are tacitly assuming that the equation is a true statement. Instead, work with the expression on the left side to show that it is equal to I.
to get A(A-1 + B-1)B = (A+B)
(A-1 + B-1)B = A-1(A+B)
A-1 + B-1 = A-1(A+B)B-1
A-1 + B-1 = (A-1A + A-1B)B-1
A-1 + B-1 = ( IB-1 + A-1I )
A-1 + B-1 = A-1 + B-1

I've gotten the same thing on both sides, and yet again I don't even know what I've done.
thanks PF

icesalmon
well, it says it is the inverse of ....
(a-1 + b-1)-1(a-1 + b-1)1 = (a-1 + b-1)0 = I
so I would say, (a-1 + b-1)-1
what do you think? Thanks you for your help.

icesalmon
If you do this, you are tacitly assuming that the equation is a true statement. Instead, work with the expression on the left side to show that it is equal to I.
I know I've done this in other threads, so it must seem like I'm not listening, I just get so careless. Thanks for the help