(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that A and Inverse(I+A) commute (where I is the identity matrix).

2. Relevant equations

Inverse(Inverse(A))=A

Inverse(AB)=Inverse(B)*Inverse(A)

3. The attempt at a solution

My solution assumes the existence of the inverse of A.

A*Inverse(I+A) = Inverse(Inverse(A))*Inverse(I+A)

= Inverse[(I+A)*Inverse(A)]

= Inverse[Inverse(A)+I]

= Inverse[Inverse(A)*(I+A)]

= Inverse(I+A)*Inverse(Inverse(A))

= Inverse(I+A)*A

My professor told me that the inverse of A may or may not exist. Does he want me to prove that it does exist? Can you even prove it does exist from the fact that the inverse of (I+A) exists? Does he want a different proof? Is he just giving me a hard time?

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# Homework Help: Matrix Multiplication and Inverses

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