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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

    EnumaElish

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    AB is idempotent ==> AB = (AB)^2. How far are you from showing this?
     
  4. Oct 15, 2007 #3
    i have to start with AB=BA and go from there to finally end up with (AB)=(AB)^2.
     
  5. Oct 15, 2007 #4

    EnumaElish

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    I'd pose it differently. You should start with the definition of idempotence as it applies to AB. Then work your way to (AB)=(AB)^2 using the information given:
    1. A is idempt.
    2. B is idempt.
    3. AB = BA.

    So, how far are you from (AB)=(AB)^2 ?
     
    Last edited: Oct 15, 2007
  6. Oct 15, 2007 #5
    ab=ab
    ab=a^2b^2
    ab=aabb
    ab=abab
    ab=(ab)^2

    got it. my mistake was starting off with ab=ba. thanks bro
     
  7. Oct 15, 2007 #6

    matt grime

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    Please say you didn't pay attention to this.

    You must start from

    AB=BA amd A^2=B^2=I to show that (AB)^2=I. This is entirely trivial if you remember that matrix multiplication is associative.
     
  8. Oct 15, 2007 #7

    learningphysics

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    But idempotent means A^2 = A... not I...
     
  9. Oct 15, 2007 #8

    learningphysics

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    Looks good to me.
     
  10. Oct 16, 2007 #9

    matt grime

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    Yes, absolutely. I got that well and truly wrong.
     
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