Matrix Multiplication and Inverses

[Random Variable]

Homework Statement

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

Homework Equations

Inverse(Inverse(A))=A

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

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?

 

[Dick]

You can't prove A^(-1) exists. It might not. Suppose A=0. The result is still true. If (I+A)^(-1) exists then you must have (I+A)*(I+A)^(-1)=(I+A)^(-1)*(I+A)=I. Isn't that enough to prove it without assuming A^(-1) exists?

 

[Random Variable]

Yes, that's all you need. Thanks.