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Math Help - Help with Elem. Linear Algebra Proof?

  1. #1
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    Help with Elem. Linear Algebra Proof?

    I've been browsing over some Linear Algebra this summer, and came across this problem in a practice book. Any help on this? It should be solvable using only elementary methods.

    Let A be an nxn matrix with characteristic polynomial

    f(t)=(-1)^{n}t^{n}+a_{n-1}t^{n-1}+...+a_{1}t+a_{0}

    Prove that A is invertible if and only if a_{0} is not equal to 0.
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  2. #2
    Super Member Gamma's Avatar
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    A matrix A is not invertible if and only if 0 is an eigenvalue of A which is if and only if \chi_A(0)=f(0)=0 if and only if a_0=0
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  3. #3
    ynj
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    Quote Originally Posted by paupsers View Post
    I've been browsing over some Linear Algebra this summer, and came across this problem in a practice book. Any help on this? It should be solvable using only elementary methods.

    Let A be an nxn matrix with characteristic polynomial

    f(t)=(-1)^{n}t^{n}+a_{n-1}t^{n-1}+...+a_{1}t+a_{0}

    Prove that A is invertible if and only if a_{0} is not equal to 0.
    if a0=0, then 0 is an eigenvalue. So there exist a nonzero vector X such that AX=0X=0. If A is invertible, then A^(-1)AX=A^(-1)0=0=X, which will make contradiction. So A is not invertible. If A is not invertible, there exists nontrivial solution for the equation groups represented by A. That is, there is a nonzero X such that AX=0=0X. So 0 is an eigenvalue. Since f(t)=(t-r1)(t-r2)...(t-rk),where ri is eigenvalue, a0=(-1)^(k)r1...rk. So a0=0
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