Prove that $(I+A^{-1})^{-1}=A(A+I)^{-1}.$

Question

Prove that $(I+A^{-1})^{-1}=A(A+I)^{-1}$ assuming that the inverse matrices exist.

My idea is to show that $(I+A^{-1})$ is the inverse matrix of $A(A+I)^{-1}$ by proving $(I+A^{-1})A(A+I)^{-1}=I$ and $A(A+I)^{-1}(I+A^{-1})=I$.

I have started with

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

but when I go on to prove this the other way around

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

I am not sure how to open up the expression.

Note that in general, If $AB = I$ then $BA = I$, so proving the other way around is not strictly necessary. That being said: if you wanted to do it that way, you could start by showing that $A(A+I)^{-1} = (A+I)^{-1}A$. — Ben Grossmann, Oct 06 '18 at 11:16

score 3 · Accepted Answer · answered Oct 06 '18 at 11:23

3

A more direct manipulation would be simpler: $$\left(I+A^{-1}\right)^{-1}=\left((A+I)A^{-1}\right)^{-1}=(A^{-1})^{-1}(A+I)^{-1}=A(A+I)^{-1}$$

answered Oct 06 '18 at 11:23

keoxkeox

1 Answers1