I know I can check if they are correct using $\displaystyle A\vec{e} = \lambda\vec{e} $
But how can I tell if Eigenvectors are cross row or cross column?
I don't exactly know what you mean by "cross row" or "cross column". The eigenvalue problem is defined by the equation you have related. In that equation, $\displaystyle \vec{e}$ is a column vector. If you were to try to formulate an eigenvalue problem using row vectors, it would look like this:
$\displaystyle \vec{e}^{\;T}A^{T}=\lambda\vec{e}^{\:T}.$
Is this what you meant?