Verify matrix identity

Question

The question:

Show that if $A$, $B$, and $A+B$ are invertible matrices with the same size, then: $$A(A^{-1}+B^{-1})B(A+B)^{-1} = I$$

I began by multiplying the first $A$:

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

and then

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

At this point I'm not sure what to do. Should I just assume $A(A+B)^{-1} = 0$, or does that not work to prove this?

Answer 1

You are wrong when you multiplied by the first $A$, note $$ A(A^{-1}+B^{-1})B = (I+AB^{-1})B =(B+AI)= (B+A) $$ Then of course $(B+A)(A+B)^{-1}=I$.

Answer 2

Multiply both sides by $A+B$: you will get $A(A^{-1}+B^{-1})B=A+B$ but the left hand side is $AA^{-1}B+AB^{-1}B=B+A$ as desired.

Answer 3

\begin{eqnarray} A(A^{-1}+B^{-1})B(A+B)^{-1}&=&[A(A^{-1}+B^{-1})]B(A+B)^{-1}=(AA^{-1}+AB^{-1})B(A+B)^{-1}\\ &=&[(I+AB^{-1})B](A+B)^{-1}=(B+AB^{-1}B)(A+B)^{-1}\\ &=&(B+A)(A+B)^{-1}=(A+B)(A+B)^{-1}=I. \end{eqnarray}