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Question about invertible matrices

  1. Dec 13, 2012 #1
    1. The problem statement, all variables and given/known data
    Show that if A, B and A + B are all invertible and the same size
    then A(A-1 + B-1)B(A + B)-1 = I
    And what does the result say about A-1 + B-1


    3. The attempt at a solution
    I start off by trying to reduce the LHS as much as I can so I multiply both sides on the right by ( A + B )
    to get A(A-1 + B-1)B = (A+B)
    (A-1 + B-1)B = A-1(A+B)
    A-1 + B-1 = A-1(A+B)B-1
    A-1 + B-1 = (A-1A + A-1B)B-1
    A-1 + B-1 = ( IB-1 + A-1I )
    A-1 + B-1 = A-1 + B-1
    :confused:
    I've gotten the same thing on both sides, and yet again I don't even know what i've done.
    thanks PF
     
  2. jcsd
  3. Dec 13, 2012 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    A direct calculation on the left gives (I+ AB-1)B(A+B)-1= (B+ A)(A+ B)-1. And, of course, matrix addition is commutative.

    Well, it says it is the inverse of ....


     
  4. Dec 13, 2012 #3

    Mark44

    Staff: Mentor

    If you do this, you are tacitly assuming that the equation is a true statement. Instead, work with the expression on the left side to show that it is equal to I.
     
  5. Dec 14, 2012 #4
    (a-1 + b-1)-1(a-1 + b-1)1 = (a-1 + b-1)0 = I
    so I would say, (a-1 + b-1)-1
    what do you think? Thanks you for your help.
     
  6. Dec 14, 2012 #5
    I know i've done this in other threads, so it must seem like i'm not listening, I just get so careless. Thanks for the help
     
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