1) For each distinct real eigenvalue $\lambda$ of a 3x3 matrix $A$, it turns out that the cross product of the transpose of any two linearly independent rows of $A-\lambda I$ gives a corresponding eigenvector (and thus easily the corresponding eigenspace, since in this case the eigenspace is an eigenline). But why does this method work?
2) I think the above may be generalisable to any 3x3 matrix with only real eigenvalues: Substitute an eigenvalue of $A$ into $A-\lambda I$. Then take the cross products of the transpose of any two pairs of rows of $A-\lambda I$. Only two possibilities exist: (a) If only one is nonzero, that gives a corresponding eigenvector and hence easily the eigenspace. (b) If both are zero, then the eigenspace is the plane orthogonal to any row of $A-\lambda I$. Is this generalisation valid, and if so, why does the method work?
3) How about for the final case whereby 2 complex eigenvalues exist??