(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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}

3. 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

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Question about invertible matrices

**Physics Forums | Science Articles, Homework Help, Discussion**