Idempotent proof

  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. (AB)^2 = ABAB = AABB = A^2B^2 = AB
     
  4. can you just switch the B and A from ABAB to get AABB?
     
  5. ABAB = A(BA)B = A(AB)B = AABB. Is that OK ?
     
    Last edited: Oct 15, 2007
  6. 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. learningphysics

    learningphysics 4,124
    Homework Helper

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

    matt grime 9,396
    Science Advisor
    Homework Helper

    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. 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. learningphysics

    learningphysics 4,124
    Homework Helper

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