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Linear Algebra help

  1. Feb 16, 2010 #1
    1. The problem statement, all variables and given/known data

    Prove that if A is an n x n matrix that is idempotent and invertible, then A=I sub n

    2. Relevant equations

    none

    3. The attempt at a solution

    I don't know how to prove this. Can anyone help me with this? Thank you
     
  2. jcsd
  3. Feb 16, 2010 #2

    Mark44

    Staff: Mentor

    What does it mean that A is idempotent? Invertible? Start with those.
     
  4. Feb 16, 2010 #3
    A is idempotent if A=A^2 and invertible if there exists an n x n matrix B such that AB=AB=I sub n.
     
  5. Feb 16, 2010 #4

    Mark44

    Staff: Mentor

    OK, then what does A2 - A equal?

    Also, AB is always equal to itself, so that doesn't do you any good.
     
  6. Feb 16, 2010 #5
    I meant to write AB=BA=I sub n.
    A^2 - A= 0
     
  7. Feb 16, 2010 #6

    Mark44

    Staff: Mentor

    Yeah, that's more like it.
    OK, now what can you do with that?
     
  8. Feb 16, 2010 #7
    (a^2)- (a^-1) i=0
     
  9. Feb 16, 2010 #8

    Mark44

    Staff: Mentor

    What do you mean by that?
     
  10. Feb 16, 2010 #9
    Since B is the inverse of A, I can write AB=I as I=AA^-1. So A^2 - AA^-1 = I
     
  11. Feb 16, 2010 #10

    Mark44

    Staff: Mentor

    No, A2 - AA-1 = A - I
    But so what?

    What I asked was whether you could do anything with A2 - A = 0?

    You said
    I still don't know how you got that or how it relates to A2 - A = 0.
     
  12. Feb 16, 2010 #11
    I could factor out A^2 - A=0, which will become A(A-1)=0
     
  13. Feb 16, 2010 #12

    Mark44

    Staff: Mentor

    OK, now you're on the right track, with a correction.

    A2 - A = 0
    <==> A(A - I) = 0

    Everything in the equation above is a matrix. You can't subtract a scalar (1) from a matrix (A), so that's the reason I wrote it this way. I has the same role in matrix multiplication that 1 has in scalar multiplication.

    There are two obvious things you can say about the equation above, and one not-so-obvious thing. For starters, what are the obvious things you can say?
     
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