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).