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Homework Help: Show that BA is an idempotent matrix

  1. Nov 14, 2012 #1
    1. The problem statement, all variables and given/known data

    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

    2. Relevant equations

    3. 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


    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?
  2. jcsd
  3. Nov 14, 2012 #2


    Staff: Mentor

    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?
  4. Nov 14, 2012 #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.
  5. Nov 14, 2012 #4


    Staff: 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
  6. Nov 14, 2012 #5
    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.
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