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Math Help - Determinant of a complex matrix

  1. #1
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    Determinant of a complex matrix

    I have been given a 3x3 complex matrix and been told to find the determinant of (A-[lamba]I), which I have calculated (and double and triple checked) to be the following:

    -lambda^3 + w^2[lambda] + u^2[lambda] + v^2[lambda] + 2uvw

    where u, v, and w are complex numbers in the matrix A. The question is this: How can I use the fact that C (the field of complex numbers) is algebraically closed to conclude that det(A-[lambda]I) = (lambda - alpha)(lambda - beta)(lambda - gamma) for some complex numbers alpha, beta, gamma?
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  2. #2
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    Quote Originally Posted by scosgurl View Post
    I have been given a 3x3 complex matrix and been told to find the determinant of (A-[lamba]I), which I have calculated (and double and triple checked) to be the following:

    -\lambda^3 + w^2 \lambda + u^2 \lambda + v^2 \lambda + 2uvw

    where u, v, and w are complex numbers in the matrix A. The question is this: How can I use the fact that C (the field of complex numbers) is algebraically closed to conclude that

    \det (A- \lambda I) = \color{red}- (\lambda - \alpha)(\lambda - \beta)(\lambda - \gamma) for some complex numbers alpha, beta, gamma?
    a better question is this: do you know what "algebraically closed" means?
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