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Let $A$, $B$, and $A + B$ be nonsingular matrices. Prove that $A^{-1} + B^{-1}$ is nonsingular by showing that

$( A^{-1} + B^{-1} )^{-1} = A( A + B )^{-1} B$ I have done progress to only knowing that $( A^{-1} + B^{-1} )^{-1} = (A + B)$ and from there am lost completely. Someone please give me pointers

J. W. Tanner
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Georgy Douglas
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  • What means $A(A^ + B^)^(-1)B$? Who are $x,y$ in $A(A^{x}+B^{y})B$? – Lucas Mar 12 '19 at 03:47
  • @Georgy Douglas: If you are happy with the given answer, then accept it, please. Otherwise your question remains open indefinitely. – Moritz Mar 12 '19 at 11:32
  • Sorry, yeah am satisfied with the answer thank you – Georgy Douglas Mar 12 '19 at 12:21
  • @GeorgyDouglas See the little grey tick mark under my answer's score? If you press that, it'll turn green, I'll get 15 reputation, and the question is considered "settled" in that you're officially happy with your answer and not actively looking for new answers. That's what Moritz was referring to by "accepting" the answer. I have more than enough reputation, but I agree in principle: if you're done with a question, it's a good idea to award a tick to your most helpful answer. – Theo Bendit Mar 14 '19 at 05:31
  • https://math.stackexchange.com/q/1505628/321264 – StubbornAtom Jul 15 '20 at 15:16

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To show that $A(A + B)^{-1} B$ is the inverse to $A^{-1} + B^{-1}$, you need to verify the definition of inverses. Specifically, we call the matrix $C$ the inverse of $A^{-1} + B^{-1}$ if $$C(A^{-1} + B^{-1}) = (A^{-1} + B^{-1})C = I.$$ If such a matrix $C$ exists, then it must be unique, hence why we can call it the inverse.

That is, we just need to show, $$A(A + B)^{-1} B(A^{-1} + B^{-1}) = (A^{-1} + B^{-1})A(A + B)^{-1} B = I.$$

This is a little daunting to do as written. First note that $$A(A + B)^{-1} B + A(A + B)^{-1} A = A(A + B)^{-1}(A + B) = AI = A,$$ hence $$A(A + B)^{-1} B = A - A(A + B)^{-1} A.$$ Similarly, $$A(A + B)^{-1} B = B - B(A + B)^{-1} B.$$ Hence, \begin{align*} (A^{-1} + B^{-1})A(A + B)^{-1} B &= A^{-1} (A(A + B)^{-1} B) + B^{-1} (A(A + B)^{-1} B) \\ &= A^{-1}(A - A(A + B)^{-1} A) + B^{-1}(B - B(A + B)^{-1} B) \\ &= I - (A + B)^{-1} A + I - (A + B)^{-1} B \\ &= 2I - (A + B)^{-1} (A + B) \\ &= I. \end{align*} See if you can show that $$A(A + B)^{-1} B(A^{-1} + B^{-1}) = I$$ in a similar manner!

Theo Bendit
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