- #1

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## 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

^{-1}A + A

^{-1}B)B

^{-1}

A

^{-1}+ B

^{-1}= ( IB

^{-1}+ A

^{-1}I )

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