Show that BA is an idempotent matrix

  • #1

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?
 

Answers and Replies

  • #2
35,303
7,178

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?
 
  • #3
(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.
 
  • #4
35,303
7,178
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
 
  • #5
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.
 

Related Threads on Show that BA is an idempotent matrix

  • Last Post
Replies
4
Views
3K
Replies
1
Views
6K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
1
Views
7K
  • Last Post
Replies
1
Views
2K
Replies
23
Views
5K
Replies
4
Views
2K
Replies
1
Views
2K
  • Last Post
Replies
7
Views
3K
  • Last Post
Replies
2
Views
48K
Top