Linear Algebra: Showing matrix equivalency.

Question

Question: If A, B, and A+B are invertible matrices, show that $$A(A+B)^{-1}B = B(A+B)^{-1}A = (A^{-1}+B^{-1})^{-1}$$

Showing the first two matrices are equivalent is simple, because I can keep on multiplying matrices until I get what I want. It's showing the equivalency to the third matrix that I'm having trouble with. How can I manipulate either of the first two into the third?

https://math.stackexchange.com/q/1505628/321264 – StubbornAtom Jan 28 '22 at 13:37 — StubbornAtom, Jan 28 '22 at 13:37

EA304GT · Accepted Answer · 2015-11-15T06:27:02.157

2

First of all, remember that for any invertible matrices $A,B$ we have the identity $(AB)^{-1}=B^{-1}A^{-1}$ which can be deduced from the fact that $I=(AB)^{-1}(AB)$. This identity can be generalized for three matrices as $(ABC)^{-1}=C^{-1}B^{-1}A^{-1}$ using the fact that $I=(ABC)^{-1}(ABC)$

Now notice the identity:

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

then factoring the $A^{-1}$ on the left side and knowing that addition is conmutative on the right side:

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

Taking the inverse on both sides:

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

edited Nov 15 '15 at 06:27

answered Nov 15 '15 at 05:48

EA304GT

1,061

Thank you! Would you mind explaining your steps a little bit? I'm having a hard time following them. – peachyloaf Nov 15 '15 at 06:19
Don't worry to ask anything else if needed :-) – EA304GT Nov 15 '15 at 06:27

Linear Algebra: Showing matrix equivalency.

1 Answers1