## idempotent proof

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?

 Quote by eyehategod 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
 can you just switch the B and A from ABAB to get AABB?

## idempotent proof

ABAB = A(BA)B = A(AB)B = AABB. Is that OK ?
 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.

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 Quote by eyehategod 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.

 Quote by eyehategod 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.
 Recognitions: Homework Help Science Advisor 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.
 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?

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 Quote by eyehategod 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...