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

  • Thread starter eyehategod
  • Start date
  • Tags
    Proof
In summary, if A and B are idempotent and AB=BA, then it can be proven that AB is also idempotent. This can be shown by using the fact that AB=BA to simplify the expression and arrive at the conclusion that (AB)^2 = AB. This also implies that A and B are invertible, but only in the trivial case. Therefore, AB is idempotent.
  • #1
eyehategod
82
0
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?
 
Physics news on Phys.org
  • #2
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
 
  • #3
can you just switch the B and A from ABAB to get AABB?
 
  • #4
ABAB = A(BA)B = A(AB)B = AABB. Is that OK ?
 
Last edited:
  • #5
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.
 
  • #6
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.
 
  • #7
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.
 
  • #8
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:
  • #9
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...
 

1. What does it mean for a matrix to be idempotent?

Idempotent matrices are those that when multiplied by themselves, result in the same matrix. In other words, the matrix A is idempotent if A^2 = A.

2. How can we prove that AB is idempotent if A=A^2?

We can prove this by using the properties of matrix multiplication. First, we multiply both sides of A=A^2 by B, which gives us AB = A^2B. Then, we can substitute A^2 with A, giving us AB = AB. Since AB is equal to itself, it is idempotent.

3. Is AB=BA also a requirement for proving idempotency?

No, AB=BA is not a requirement for proving idempotency. It is only a requirement if we want to prove that A and B commute, which means they can be multiplied in any order and still result in the same product.

4. Can a matrix be idempotent if it is not a square matrix?

Yes, a matrix can be idempotent even if it is not a square matrix. As long as A and B are both m x n matrices, the equation AB=BA can still hold true, making AB an idempotent matrix.

5. Are there any real-world applications of idempotent matrices?

Yes, idempotent matrices have various applications in fields such as economics, physics, and computer science. In economics, they are used to model stable systems, while in physics, they are used to describe systems that do not change over time. In computer science, they are used in algorithms and data structures.

Similar threads

  • Linear and Abstract Algebra
Replies
1
Views
613
  • Linear and Abstract Algebra
Replies
15
Views
2K
  • Precalculus Mathematics Homework Help
Replies
18
Views
2K
  • Calculus and Beyond Homework Help
2
Replies
40
Views
3K
  • Linear and Abstract Algebra
Replies
1
Views
798
  • Linear and Abstract Algebra
Replies
1
Views
1K
  • Linear and Abstract Algebra
Replies
6
Views
2K
  • Linear and Abstract Algebra
Replies
2
Views
969
  • Calculus and Beyond Homework Help
Replies
2
Views
2K
  • Linear and Abstract Algebra
Replies
8
Views
2K
Back
Top