Eigen values and characteristic equation closed

Question

I have learned a shortcut for finding eigen values and characteristic equation. It is as follows.

$\lambda ^{_{3}}- \alpha \lambda ^{2}+ \beta \lambda - \gamma =0$

where $ \alpha =$ trace of 3*3 matrix

$ \beta =$ sum of cofactor of diagonal elements

$ \gamma =$ det of 3*3 matrix

when I follow this logic in the below problem  I'm stuck,  Could any one help me find why I'm stuck .  I was getting answer using this trick for almost all problems

Comments

in beta,

your last minor , i.e minor obtained after removing 3rd row & 3rd col is wrong

it should be  [ 1    1 ]

                  [  0    2 ]