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Idempotent proof

  1. Oct 15, 2007 #1
    If A and B are idempotent(A=A^2) and AB=BA, prove that AB is idempotent.

    this is what i got so far.
    AB=BA
    AB=B^(2)A^(2)
    AB=(BA)^(2)

    this is where I get stuck.
    Do A and B have inverses? if so, why?
    should I be thinking about inverses or is there another way of approaching this problem?
     
  2. jcsd
  3. Oct 15, 2007 #2
    (AB)^2 = ABAB = AABB = A^2B^2 = AB
     
  4. Oct 15, 2007 #3
    can you just switch the B and A from ABAB to get AABB?
     
  5. Oct 15, 2007 #4
    ABAB = A(BA)B = A(AB)B = AABB. Is that OK ?
     
    Last edited: Oct 15, 2007
  6. Oct 15, 2007 #5
    i doubt you can switch those matrices b/c youre multiplying the two of them. This has got be wrong. there has to be a different way to get what im trying to get.
     
  7. Oct 15, 2007 #6

    learningphysics

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    It is given in the question that AB = BA... so it's ok to switch them.
     
  8. Oct 15, 2007 #7
    YOU said in the first post AB=BA, IF that is true then you can switch the order like that.
     
  9. Oct 15, 2007 #8

    matt grime

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    As has been pointed out THEY COMMUTE! But that isn't why I post. I want to point out that only in the trivial case can an idempotent be invertible.
     
    Last edited: Oct 15, 2007
  10. Oct 15, 2007 #9
    so if i start the proof off with AB=BA then I can use AB=BA later on in the proof I started off with in the firszt place?
     
    Last edited: Oct 15, 2007
  11. Oct 15, 2007 #10

    learningphysics

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    Yes. you're given AB = BA is true... so you can use that anywhere in your proof...
     
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