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Math Help - Theory of Matrices

  1. #1
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    Theory of Matrices

    Let A belonging M_n be a non-singular matrix. Let p_A(t) be its characteristic polynomial, p_A(t) = det(tI-A) = t^n + a_(n-1)t^(n-1) + ... + a_1t + a_0.

    a) Prove that a_0 = (-1)^n det(A)
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  2. #2
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    Re: Theory of Matrices

    Quote Originally Posted by page929 View Post
    Let A belonging M_n be a non-singular matrix. Let p_A(t) be its characteristic polynomial, p_A(t) = det(tI-A) = t^n + a_(n-1)t^(n-1) + ... + a_1t + a_0.

    a) Prove that a_0 = (-1)^n det(A)

    a_0=p_A(0)=\det (0I-A)=\det(-A) . Now, note that -A is the result of changing the sign to all columns of A.

    P.S. The hypothesis A non-singular is irrelevant.
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    Re: Theory of Matrices

    By the way, to get all of a subscript or all of a superscript above or below the main level, in LaTeX, put them in { }: e^{ax_{1+1}} gives e^{ax_{1+1}}.
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    Re: Theory of Matrices

    Quote Originally Posted by FernandoRevilla View Post
    a_0=p_A(0)=\det (0I-A)=\det(-A) . Now, note that -A is the result of changing the sign to all columns of A.
    or, det(-A) = det[(-I)(A)] = det(-I)xdet(A) = ((-1)^n)xdet(A)
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