What is the problem?

You can simplify the system to:

(By adding the last row to the middle-row)

-x-8y+4z = 0

3x = 0

2x -2y+z = 0

It follows that x = 0, and we can write the system as:

x= 0,

-8y+4z = 0

-2y+z = 0

Last 2 eqations are equivalent and we get: x=0, z = 2y so v=(0,1,2)

Is an eigenvector by lambda= 1

(I see in the last row you put a -3 for the second entry: this must be a -2)