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?