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A is square, A^2 = A -> det(A) = 0 or 1

  1. Aug 8, 2011 #1
    A is square, A^2 = A ----> det(A) = 0 or 1

    1. The problem statement, all variables and given/known data

    I need to prove the thread title.

    2. Relevant equations

    The only relevant equation I can think of is
    det(A) = 0 <---> A * inv(A) = I

    3. The attempt at a solution

    Been toying around with it but not getting anywhere. Help, brahs?
     
  2. jcsd
  3. Aug 8, 2011 #2

    Dick

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    Re: A is square, A^2 = A ----> det(A) = 0 or 1

    Your 'only relevant equation' makes no sense. Can't you do a little better than that? How is det(AB) related to det(A) and det(B)?
     
  4. Aug 8, 2011 #3
    Re: A is square, A^2 = A ----> det(A) = 0 or 1

    The determinant of a matrix A is zero if and only if there exists a matrix B such that A*B = I. In that case we call B in the inverse of A, and we denote it A-1.
     
  5. Aug 8, 2011 #4

    Dick

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    Re: A is square, A^2 = A ----> det(A) = 0 or 1

    If the determinant of a square matrix is zero, then it has no inverse. Check your book again.
     
  6. Aug 8, 2011 #5
    Re: A is square, A^2 = A ----> det(A) = 0 or 1

    hahah sorry, brah

    I took this class over 2 years ago. My buddy asked me this question and I guess I remembered some facts incorrectly.

    So det(A) = 0 iff A has no inverse. In that case I'll need another relevant equation with which I can launch into a rigorous and elegant proof. Got a suggestion, brah?
     
  7. Aug 8, 2011 #6

    Dick

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    Re: A is square, A^2 = A ----> det(A) = 0 or 1

    Look at my post #2. Product of matrices is related to product of determinants, isn't it?
     
  8. Aug 9, 2011 #7
    Re: A is square, A^2 = A ----> det(A) = 0 or 1

    It's not obvious to me that it is. Prove it.
     
  9. Aug 9, 2011 #8

    HallsofIvy

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    Re: A is square, A^2 = A ----> det(A) = 0 or 1

    det(AB)= det(A)det(B) is true but I would not use it. If [itex]A^2= A[/itex], then [itex]A^2- A= A(A- I)= 0[/itex]. So either A or A- I is not invertible.
     
  10. Aug 9, 2011 #9

    Ray Vickson

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    Re: A is square, A^2 = A ----> det(A) = 0 or 1

    Here are some hints.
    (1) Go to the library.
    (2) Get a book.
    (3) Read it.

    RGV
     
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