# 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?

## Answers and Replies

Related Linear and Abstract Algebra News on Phys.org
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?

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

Last edited:
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.

learningphysics
Homework Helper
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.

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.

matt grime
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:
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:
learningphysics
Homework Helper
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...