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

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The discussion centers on proving that the product of two idempotent matrices, A and B, is also idempotent under the condition that AB = BA. The participants establish that if A and B are idempotent (A = A² and B = B²), then the equation (AB)² = AB holds true, confirming that AB is idempotent. Key points include the ability to switch the order of multiplication due to the commutative property of A and B and the clarification that idempotent matrices are not generally invertible except in trivial cases.

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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?
 
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eyehategod said:
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 I am trying to get.
 
eyehategod said:
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.
 
eyehategod said:
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.
 
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:
  • #10
eyehategod said:
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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