Originally Posted by

**scosgurl** 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:

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

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

$\displaystyle \det (A- \lambda I) = \color{red}-$$\displaystyle (\lambda - \alpha)(\lambda - \beta)(\lambda - \gamma)$ for some complex numbers alpha, beta, gamma?