I have this matrix:

2 1 -1 0

0 4 -2 0

0 3 -1 0

0 3 -2 1

I need to find the eigenvalues.

So I get the matrix

(x-2) 1 -2 1

0 (x-4) -4 2

0 3 (x+4) 3

0 3 -5 (x-4)

I take the determinant:

(x-2)*[(x-4)(x+4)(x-4)-36-30-6(x+4)+15(x-4)+12(x-4)]=

(x-2)*[(x-4)(x+4)(x-4)-66-6(x+4)+27(x-4)]

Clearly, 2 is an eigenvalue. I get messy other ones though so it doesn't seem right. How would you solve a complex cubic equation anyway?

Can anyone check this?