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Homework Help: Inverse of a linear operator

  1. Feb 2, 2009 #1
    1. The problem statement, all variables and given/known data

    Show that if an operator A satisfies A2 - A + I = 0 then A has an inverse. Express A-1 as a simple polynomial of A.

    2. Relevant equations

    I'm not sure that this is relevant, but A-1=1/(detA)TrC where TrC is the transpose of the matrix of cofactors. Also:
    If detA = 0 then the matrix has no inverse

    3. The attempt at a solution
    So I notice immediately that adding by the identity matrix in this equation will result in a matrix with its diagonal having numbers (real or complex) and the rest being zero, as I can be expressed as the kronecker delta. And if the determinant must be nonzero in order to have an inverse, there has to be a way to relate the diagonal of an n dimensional matrix with its determinant. I'm just stuck as to how to do that. Any help greatly appreciated, thank you.
    I've been thinking more about this problem, and it seems like there should be a way to use the secular equation to solve it. We went over it briefly in class (the class is quantum and I haven't had linear algebra yet, so it's kind of a chore), but not in enough detail that I would be able to use it in a proof.
    Last edited: Feb 2, 2009
  2. jcsd
  3. Feb 2, 2009 #2


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    Welcome to PF!

    Hi seek! Welcome to PF! :smile:
    oooh … so complicated:cry:

    Try writing it A2 - A = -I :wink:
  4. Feb 3, 2009 #3
    My oversight is to my pride as a cold slap to the visage. Thanks so much for the help, maybe next time I'll be able to use the skills I learned in 5th grade.
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