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Simple Linear Algebra Proof

  1. Sep 12, 2009 #1
    1. The problem statement, all variables and given/known data
    Prove that given a matrix A, and A^2 = A, then A must be either the zero matrix or the identity matrix.

    3. The attempt at a solution
    By multiplying both sides by A, you can deduce that A = A^2 = A^3 = A^4 ...
    From there I think it's obvious that A must be either 0 or I, but I don't know how to start proving it formally.

    Thanks very much for your help
    -Patrick
     
  2. jcsd
  3. Sep 12, 2009 #2
    Try taking the determinant of A
     
  4. Sep 12, 2009 #3
    Is there a way to prove it without using the determinant. This exercise is given before determinants are introduced.

    Thanks
    -Patrick
     
  5. Sep 12, 2009 #4

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    If [itex]A^2= A[/itex], then [itex]A^2- A= A(A- I)= 0[/itex].

    However, you have to be careful here. With matrices it is NOT true that "if AB= 0 then either A= 0 or B= 0".
     
  6. Sep 12, 2009 #5
    Is that true? What about A = [1,0;0,0]? This is not zero or I but A^2 = A
     
  7. Sep 12, 2009 #6
    Ah. I didn't spot that buzzmath. Thank you. The question actually does say either prove or find a counterexample. I was just too sure that it was true.

    Thanks
    -Patrick
     
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