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Homework Help: Linear Algebra - invertible matrix; determinants

  1. Nov 10, 2008 #1
    1. The problem statement, all variables and given/known data

    Prove that
    [1 a b
    -a 1 c
    -b -c 1]
    is invertible for any real numbers a,b,c

    2. Relevant equations

    A is invertible if and only if det[A] does not equal 0.

    3. The attempt at a solution

    I'm not sure if I'm going about this in the correct way;
    Would I prove this by solving for the determinant? I did this by cofactor expansion, and came up with:
    (1+c^2) - a(-a+bc) + b(ac+b)
    = a^2 + b^2 + c^2 + 1

    Could I just say then, that the determinant could never be zero, since
    a^2 + b^2 + c^2 + 1
    will always be nonzero for any real numbers a,b,c?

    If someone could just let me know if I did this correctly, or if there is more I need to show.

    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Nov 10, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    Looks like you got it completely right. :cool:
    In fact you have shown that always det(A) > 1.
  4. Nov 10, 2008 #3
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