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Random Variable
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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?