6.1. EIGENVALUES AND EIGENVECTORS OF A MATRIX 147

and the row reduced echelon form is 1 0 −1 0

0 1 0 0

0 0 0 0

Therefore, the eigenvectors are of the form

z

 1

0

1

where z ̸= 0.

Next find the eigenvectors for λ = 2. The augmented matrix for the system of equationsneeded to find these eigenvectors is 0 −2 2 0

−1 −1 1 0

1 −1 1 0

and the row reduced echelon form is 1 0 0 0

0 1 −1 0

0 0 0 0

and so the eigenvectors are of the form

z

 0

1

1

where z ̸= 0.

Finally find the eigenvectors for λ = 4. The augmented matrix for the system of equationsneeded to find these eigenvectors is 2 −2 2 0

−1 1 1 0

1 −1 3 0

and the row reduced echelon form is 1 −1 0 0

0 0 1 0

0 0 0 0

 .

Therefore, the eigenvectors are of the form

y

 1

1

0

where y ̸= 0.