Idempotent proof

  • Thread starter eyehategod
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  • #1
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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?
 

Answers and Replies

  • #2
229
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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?
(AB)^2 = ABAB = AABB = A^2B^2 = AB
 
  • #3
82
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can you just switch the B and A from ABAB to get AABB?
 
  • #4
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ABAB = A(BA)B = A(AB)B = AABB. Is that OK ?
 
Last edited:
  • #5
82
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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.
 
  • #6
learningphysics
Homework Helper
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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.
It is given in the question that AB = BA... so it's ok to switch them.
 
  • #7
1,074
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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.
YOU said in the first post AB=BA, IF that is true then you can switch the order like that.
 
  • #8
matt grime
Science Advisor
Homework Helper
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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:
  • #9
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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:
  • #10
learningphysics
Homework Helper
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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?
Yes. you're given AB = BA is true... so you can use that anywhere in your proof...
 

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