# Determinant of a complex matrix

• Apr 26th 2009, 07:49 PM
scosgurl
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?
• Apr 26th 2009, 07:59 PM
NonCommAlg
Quote:

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:

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

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?