I have the matrix

2 1 -1 0

0 4 -2 0

0 3 -1 0

0 3 -2 1

The eigenvectors are 1,1,2,2.

To get the eigenvectors for 1,1, I use the matrix

-1 1 -1 0

0 -3 -2 0

0 3 2 0

0 3 -2 0

When I bring this to RREF, I have

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 0

So I know that the vector

0

0

0

1

is an eigenvector.

But the book also says that

1

2

3

0

is an eigenvector for the eigenvalue 1.

If the matrix only has one solution, how do I get this other eigenvector?