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Linear Algebra

  1. Feb 23, 2010 #1
    1. The problem statement, all variables and given/known data

    Let A be an idempotent matrix (A^2=A)
    a) show that I-A is also idempotent
    b) show that I+A is nonsingular and (I+A)^(-1)=I-(1/2)*A


    2. Relevant equations



    3. The attempt at a solution

    I managed to solve part a, but have a little problem solving b.

    I tried
    (I+A)*(I+A)^(-1)=(?)I

    but didn't know how to treat (I+A)^(-1)

    Thanks,
    Roni.
     
  2. jcsd
  3. Feb 24, 2010 #2

    Mark44

    Staff: Mentor

    You are given that A2 = A.
    Adding I to both sides gives us
    A2 + I = A + I
    ==> |A2 + I| = |A + I|

    Can you do anything with that?
     
  4. Feb 24, 2010 #3
    oh, nice approach.
    Thanks for the help.
    I'll try the second part of B
     
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