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Math Help - eigenvalues of A inverse

  1. #1
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    eigenvalues of A inverse

    I need to prove the following: If matrix A is nonsingular, then the eigenvalues of A inverse are the reciprocals of the eigenvalues of A.

    Here is what I know:

    det(A-lambdaI)=0

    det(A inverse)=1/det(A)

    I feel like I need to use this second fact somehow in my proof, but I am struggling with how to get to this step.

    Help!
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  2. #2
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    Quote Originally Posted by ecc5 View Post
    I need to prove the following: If matrix A is nonsingular, then the eigenvalues of A inverse are the reciprocals of the eigenvalues of A.

    Here is what I know:

    det(A-lambdaI)=0

    det(A inverse)=1/det(A)

    I feel like I need to use this second fact somehow in my proof, but I am struggling with how to get to this step.

    Help!

    Av=\lambda v
    A^{-1}(Av)=A^{-1}(\lambda v)=\lambda A^{-1}v
    v = \lambda A^{-1}v
    A^{-1}v=\lambda^{-1}v

    Tonio
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  3. #3
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    Oh!! Thank you! I was definitely overthinking it.
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