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Idempotent proof

by eyehategod
Tags: idempotent, proof
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eyehategod
#1
Oct15-07, 11:35 AM
P: 85
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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ircdan
#2
Oct15-07, 12:37 PM
P: 229
Quote Quote by eyehategod View Post
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
#3
Oct15-07, 01:27 PM
P: 85
can you just switch the B and A from ABAB to get AABB?

Sourabh N
#4
Oct15-07, 01:53 PM
P: 633
Idempotent proof

ABAB = A(BA)B = A(AB)B = AABB. Is that OK ?
eyehategod
#5
Oct15-07, 02:02 PM
P: 85
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
#6
Oct15-07, 02:31 PM
HW Helper
P: 4,124
Quote Quote by eyehategod View Post
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.
d_leet
#7
Oct15-07, 02:31 PM
P: 1,075
Quote Quote by eyehategod View Post
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
#8
Oct15-07, 02:32 PM
Sci Advisor
HW Helper
P: 9,396
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
#9
Oct15-07, 02:42 PM
P: 85
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
#10
Oct15-07, 03:02 PM
HW Helper
P: 4,124
Quote Quote by eyehategod View Post
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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