# Idempotent proof

1. Oct 15, 2007

### 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?

2. Oct 15, 2007

### EnumaElish

AB is idempotent ==> AB = (AB)^2. How far are you from showing this?

3. Oct 15, 2007

### eyehategod

i have to start with AB=BA and go from there to finally end up with (AB)=(AB)^2.

4. Oct 15, 2007

### EnumaElish

I'd pose it differently. You should start with the definition of idempotence as it applies to AB. Then work your way to (AB)=(AB)^2 using the information given:
1. A is idempt.
2. B is idempt.
3. AB = BA.

So, how far are you from (AB)=(AB)^2 ?

Last edited: Oct 15, 2007
5. Oct 15, 2007

### eyehategod

ab=ab
ab=a^2b^2
ab=aabb
ab=abab
ab=(ab)^2

got it. my mistake was starting off with ab=ba. thanks bro

6. Oct 15, 2007

### matt grime

Please say you didn't pay attention to this.

You must start from

AB=BA amd A^2=B^2=I to show that (AB)^2=I. This is entirely trivial if you remember that matrix multiplication is associative.

7. Oct 15, 2007

### learningphysics

But idempotent means A^2 = A... not I...

8. Oct 15, 2007

### learningphysics

Looks good to me.

9. Oct 16, 2007

### matt grime

Yes, absolutely. I got that well and truly wrong.

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