Eigenvalues for 3x3 Matrix

Trying to find eigenvalues for this matrix $\displaystyle A - \lambda I$

$\displaystyle \left(\begin{array}{ccc}{1 - \lambda} & 2 & {-2} \\ 2 & {1 - \lambda} & 0 \\ {-2} & 0 & {1 - \lambda} \end {array} \right)$

If my math is correct, the determinant of this is $\displaystyle {(1-\lambda)}^3 - 8\lambda - 12$, but this doesn't factor, so I don't know how to find the roots.

Can somebody help? Thanks.

Re: Eigenvalues for 3x3 Matrix

your math is incorrect.

the determinant is:

(1 - λ)^{3} - (-2)(-2)(1 - λ) - (2)(2)(1 - λ) = 1 - 3λ + 3λ^{2} - λ^{3} - 8(1 - λ) = -7 +5λ + 3λ^{2} - λ^{3}

this has one rational root, and two irrational roots which are "conjugate" (in the sense of square roots).