Note... I am fairly certain that my characteristic equation is correct. I just need someone to look at the eigenvectors to tell me if I have them correct. I included the matrix A, the lambdas and the rref for background. Thanks. I had an original 3x3 matrix A of:

[1 2 -1]

[2 1 1]

[-1 1 0]

for lambda 3 i get

[-2 2 -1]

[2 -2 1 ]

[-1 1 -3] rref to:

[1 -1 0]

[0 0 1]

[0 0 0]

so x - y = 0

z = 0

vector is

[1]

[1]

[0]

lamda = -2

[-3 2 -1 ]

[2 -3 1]

[-1 1 2]

rref

[1 0 0]

[0 1 0]

[0 0 1]

i get for the vector

[0]

[0]

[0]

lambda = 1

[0 2 -1]

[2 0 1]

[-1 1 -1]

rref

[1 0 1/2]

[0 1 -1/2]

[0 0 0]

let z = t

R1: x = -1/2t

R2: y = 1/2t

pick t = 2

[-1]

[1]

[2]