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Homework Help: A question in finding invertion

  1. Mar 7, 2008 #1
    i am given an operator
    S:R3[x]->R3[x]

    and we have the polinomial from which we take the eigenvalues from
    t^3 - 4t
    find wether S invertable or not???


    i tried to think about that and i got that the aigenvalues are 2 , 0, -2
    but that only prooves that it diagonizable

    i know a law that if the determinant differs zero then its invertable
    but i dont know how to apply it here
    ???
     
  2. jcsd
  3. Mar 7, 2008 #2

    Dick

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    If it has a zero eigenvalue then it has a nontrivial kernel. It's not invertible.
     
  4. Mar 8, 2008 #3

    HallsofIvy

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    Hopefully, you know that works both ways because it is the other way you need here: If the determinant of a matrix is 0, then it is not invertible.

    When you diagonalize a matrix, its determinant stays the same. The determinant of a diagonal matrix is the product of the numbers on its diagonal. Since the eigenvalues are 2, 0, and -2, the "diagonalized" matrix would have those numbers on its diagonal and so its determinant is 0.

    Of course, Dick's method works just as well.
     
  5. Mar 8, 2008 #4
    ahhh because its determinant is the multiplication of the diagonal
    and we got a zero in there
    then the whole thing is zero

    thanks
     
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