Can anyone solve the attached file?

The given matrix is


A given vector is of the form [a b c]'.
If a given vector {x} is an eigenvector of A, then

 
Test the vector x₁ = [1 -2 1]'

This is not an eigenvector because the determinant is not zero.

Test x₂ = [-1 0 1]'

This is not an eigenvector because the determinant is not ero.

Test x₃ = [0 0 0]'

This vector is an eigevector because the determinant is zero.

The eigenvector is x₃