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Math Help - Prove, about characteristic polynomial...

  1. #1
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    Prove, about characteristic polynomial...

    If A is a square matrix of order n, then prove that
    det(\lambda I-A)=\lambda ^n-tr(A)\lambda ^{n-1}+P_{n-2}(\lambda)
    with P_{n-2} is the polynomial of order (teh greatest) n-2\text{ in }\lambda
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  2. #2
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    Quote Originally Posted by GOKILL View Post
    If A is a square matrix of order n, then prove that
    det(\lambda I-A)=\lambda ^n-tr(A)\lambda ^{n-1}+P_{n-2}(\lambda)
    with P_{n-2} is the polynomial of order (teh greatest) n-2\text{ in }\lambda

    Follows at once from the definition of char. polynomial and the definition of matrix determinant: if A=(a_{ij}) , then |\lambda I-A|=\left|\begin{pmatrix}\lambda-a_{11}&\ldots & -a_{1n}\\ \ldots &\ldots &\ldots\\-a_{n1}&\ldots &\lambda-a_{nn}\end{pmatrix}\right|.

    Well, check what the coefficient of \lambda^{n-1} is above: precisely the same as in (\lambda-a_{11})\cdot\ldots\cdot(\lambda-a_{nn}) , i.e. -tr.(A) .

    Tonio
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