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Thread: Theory of Matrices

  1. #1
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    Theory of Matrices

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

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

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

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

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

    P.S. The hypothesis $\displaystyle A$ 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 $\displaystyle e^{ax_{1+1}}$.
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    Re: Theory of Matrices

    Quote Originally Posted by FernandoRevilla View Post
    $\displaystyle a_0=p_A(0)=\det (0I-A)=\det(-A)$ . Now, note that $\displaystyle -A$ 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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