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Homework Help: Matrix Multiplication and Inverses

  1. Jan 23, 2009 #1
    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?
     
  2. jcsd
  3. Jan 23, 2009 #2

    Dick

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    Science Advisor
    Homework Helper

    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?
     
  4. Jan 23, 2009 #3
    Yes, that's all you need. Thanks.
     
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