# Show that BA is an idempotent matrix

## 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

## 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?

Mark44
Mentor

## 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

## 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.

Mark44
Mentor
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

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.