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Linear Algebra - Proof of the characteristic polynomial for the inverse of A

  1. Nov 23, 2009 #1
    1. The problem statement, all variables and given/known data

    A is an nxn invertible matrix. Show that

    PA-1(x) = ([tex]\frac{(-x)^n}{det(A)}[/tex]) *PA(1/X)

    2. Relevant equations
    det(AB) = det(A)det(B)


    3. The attempt at a solution

    Ok from what I understand this is the proof for the characteristic polynomial for the inverse of a matrix. The professor said that we need to use somehow the equation above.

    I also understand that the equation is usually of the form det(A-[tex]\lambda[/tex]I) = 0, but I cannot see how to form this given the final equation which we need to produce.
     
  2. jcsd
  3. Nov 22, 2010 #2
    Try substituting the definition of the characteristic polynomial into the equation you are trying to verify. You get [tex] \det(Ix- A^{-1}) = \frac{(-x)^n\det(Ix^{-1} - A)}{\det(A)} [/tex] If you notice that [tex] \det(-Ix) = (-x)^n [/tex], then the rest is matrix arithmetic using the equation you provided.

    Good luck!
     
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