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Thread: Cofactor Expansion

  1. #1
    Sep 2008

    Cofactor Expansion

    Let A be an n x n matrix.
    a) Show that f(t) = det(tIn - A) is a polynomial in t of degree n.
    b) What is the coefficient of t^n in f(t)?
    c) What is the constant term in f(t)?

    I don't understand how I would show it in a simpler way, than the way
    that I am currently doing. I just need feedback about the way I'm
    solving the problem!

    I know that tIn would be an identity matrix multiplied by t. If I
    subtract A from tIn than I would get a matrix like this:

    |t - a11 ... -a1n|
    |-a21 ...... -a2n|
    |-an1 ... t - ann|

    For the first part, I understand how the determinant is a polynomial
    in the form of t^n.
    The first part of the determinant that would be calculated would be
    the diagonal, (a11)(a22)(a33)..(ann). That would give me the equation
    The resulting equation would end in
    t^n +- .... +- a11a22a33..ann. My first question is, is there an
    easier way to prove this, or is this good enough.

    For part b, I'm not sure if I was supposed to come up with a different
    coefficient, besides 1. I did some examples with n = 3,4,5 and I
    always got that the coefficient was 1.

    For part c I got that the constant from part a (+- a11a22..ann) would
    be added to the opposite diagonal (+-a1n a2n-1...an1).

    I'm not sure if I did that right! Thanks so much for your help!
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  2. #2
    Global Moderator

    Nov 2005
    New York City
    By proving this by induction on $\displaystyle n$ when you expand along on of the cofactors.
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