Show that BA is an idempotent matrix

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Homework Statement



A square matrix P is called an idempotent if P^2 = P,

Show that if A is n x m and B is m x n, and if AB = In, then BA is an idempotent

Homework Equations


The Attempt at a Solution



I have this one identity I think, but I'm not sure how to use it?

If A is m x n, then ImA = A = AIn

B*In*In*A

can I do anything with that?

My inverse definition shows for square matrices... how does it apply for non-square matrices? Or how can I use it?
 
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Incognitopad said:

Homework Statement



A square matrix P is called an idempotent if P^2 = P,

Show that if A is n x m and B is m x n, and if AB = In, then BA is an idempotent


Homework Equations





The Attempt at a Solution



I have this one identity I think, but I'm not sure how to use it?

If A is m x n, then ImA = A = AIn

B*In*In*A

can I do anything with that?

My inverse definition shows for square matrices... how does it apply for non-square matrices? Or how can I use it?

A and B aren't square, so they don't have inverses.

Think about what you need to do, which is to show that BA is idempotent. What does that mean in terms of the definition?
 
(BA)^2 = (BA)(BA) = BABA

once again, WOW that was easy

= B(AB)A
= B(In)A
= B(In*A)
= B(A)
= BA

thanks mark.
 
The basic idea in these types of proofs is to replace the words in the problem with their definitions.

To show: BA is idempotent
Translation: (BA)(BA) = BA
 
Mark44 said:
The basic idea in these types of proofs is to replace the words in the problem with their definitions.

To show: BA is idempotent
Translation: (BA)(BA) = BA

yeah, its just originally without writing it out i assumed (BA)^2 was equal to B^2 * A^2

need to get used to matrix multiplication.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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