Proving AB is Idempotent: A=A^2, AB=BA

[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?

 

[ircdan]

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

 

[eyehategod]

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

 

[Sourabh N]

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

 

[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 I am trying to get.

 

[learningphysics]

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 I am trying to get.

It is given in the question that AB = BA... so it's ok to switch them.

 

[d_leet]

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 I am trying to get.

YOU said in the first post AB=BA, IF that is true then you can switch the order like that.

 

[matt grime]

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.

 

[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?

 

[learningphysics]

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...