The given matrix is
[tex] A= \left[\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right] [/tex]
A given vector is of the form [a b c]'.
If a given vector {x} is an eigenvector of A, then
[tex]det \left[\begin{array}{ccc}1-a&2&3\\4&5-b&6\\7&8&9-c\end{array}\right] =0[/tex]
Test the vector x₁ = [1 -2 1]'
[tex]det \left[\begin{array}{ccc}0&2&3\\4&7&6\\7&8&8\end{array}\right] =-2(32-42)+3(32-49)=-31[/tex]
This is not an eigenvector because the determinant is not zero.
Test x₂ = [-1 0 1]'
[tex] det \left[\begin{array}{ccc}2&2&3\\4&5&6\\7&8&8\end{array}\right] =2(40-48)-2(32-42)+3(32-35)=-5[/tex]
This is not an eigenvector because the determinant is not ero.
Test x₃ = [0 0 0]'
[tex]det \left[\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right] =(45-48)-2(36-42)+3(32-35)=0[/tex]
This vector is an eigevector because the determinant is zero.
Answer:
The eigenvector is x₃
