eigenvector formula

One of them is minus the other. But it doesn't really matter as what you're most concerned about is when p(lambda)=0. Since one is minus the other then this will happen for the same values of lambda regardless of which formula you're using.

For completeness, note that if A and I are n x n matrices, then $\det(A-\lambda I) = (-1)^n \det (\lambda I - A)$. In other words one is only the minus of the other when n is odd.

For completeness, note that if A and I are n x n matrices, then $\det(A-\lambda I) = (-1)^n \det (\lambda I - A)$. In other words one is only the minus of the other when n is odd.