# How to show that if $A, B$, and $A + B$ are invertible matrices with the same size, then $A(A^{ −1} + B^{ −1} )B(A + B)^ {−1} = I$ [duplicate]

1. Show that if $A$, $B$, and $A + B$ are invertible matrices with the same size, then $$A(A^{-1} + B^{-1})B(A + B)^{-1} = I$$
2. What does the result in part $1$ tell you about the matrix $A^{-1} + B^{-1}$?

Ok I never came across any identities that would allow me to cancel these values out in the book so far... So I have no idea how I'm supposed to "show" that they are equal... So how do I solve/show this?

Note: for those who say this is a duplicate, read both questions again, this question asks how to solve the first part, the other question asks and only answers how to solve the second part. the first part isn,t explained in the other question, only the second part.

• yeah.. but it doesnt really answer..
– J L
Jun 16, 2014 at 12:55
• @JL The answer is right there. Jun 16, 2014 at 12:55
• it answers the second part but not the first part
– J L
Jun 16, 2014 at 12:56
• How come? The identity tells you that $(A^{-1}+B^{-1})B(A+B)^{-1}A=I$ (multiply by $A^{-1}$ from left and then by $A$ from right). Jun 16, 2014 at 12:56
• Oh, ok: $A(A^{-1}+B^{-1})B=A+B$, does it help? Jun 16, 2014 at 12:57

$$A(A^{-1}+B^{-1})B(A+B)^{-1} = I \iff (A^{-1}+B^{-1}) = A^{-1}B^{-1}(A+B) \\\iff A^{-1}+B^{-1} = B^{-1}+A^{-1}$$