What is the Correct Way to Find Eigenvalues and Eigenvectors of a Matrix?

[Seon]

Find the eigenvalues and corresponding eigenvector of the matrix.
A=
[-4 4 8 ]
[0 0 -10]
[0 0 2 ]

[1 -1 0]
~ [0 0 1 ]
[0 0 0 ]

I calculated by A = -$\lambda$I

So,

[1-lamda -1 0 ]
[0 -lamda 1]
[0 0 -lamda]

so, lamda = 0,0, and 1

So I got

1st eigen value: 0 eigen vector (1,1,0)
2nd eigen value: 0 eigen vector (1,1,0)
3rd eigen value: 1 eigen vector (1,0,0)

1st and 2nd values were right, but third one was wrong.
I tried several times, and I always get 1(1,0,0)

What do i need to do ?
thanks

 

[mathwonk]

if you reduce the matrix, you change the eigenvalues, except for 0. don't reduce the matrix, find the characteristic polynomial of the original A.