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Matrix determinant question

  1. Apr 13, 2009 #1
    1. The problem statement, all variables and given/known data
    If A is an idempotent matrix (A^2 = A), find all possible values of det(A).


    2. Relevant equations



    3. The attempt at a solution
    I'm not sure if this is the proper way to show it, but here's what I did:

    Since A = A^2, det(A)=det(A^2)
    So det(A) = det(A)*det(A)

    Considering two cases:
    If det(A) is not 0, then
    det(A) = 1.

    The only other way to satisfy det(A) = det(A)*det(A)
    is if det(A) = 0.

    So det(A) is either 1 or 0.
     
  2. jcsd
  3. Apr 13, 2009 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Sure. If det(A)=x then you have x^2=x so x^2-x=0 so x*(x-1)=0. And, yes, that means x=0 or x=1.
     
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