Gaussian Elimination and Eigenvectors

Question

I've read in many places that Gaussian Elimination cannot be used to find the eigenvectors of a matrix. I don't understand why.

Assume we have the matrix $\mathbf{A}$ and we know the eigenvalues $\lambda$. As far as I know:

1. The eigenspace corresponding to a given eigenvalue is the nullspace of the matrix $\mathbf{A-\lambda\,\mathbf{I}}$

2. Gaussian elimination preserves the nullspace of a matrix

Given that these statements are correct (please correct me if I'm wrong) doesn't it mean that if I apply GE on $\mathbf{A-\lambda\,\mathbf{I}}$, the resulting reduced matrix must have the same nullspace as that of $\mathbf{A-\lambda\,\mathbf{I}}$ and, hence, gives the same set of eigenvectors?

What am I missing?