In this paper, we consider a first order perturbation $A(x)\cdot D+q(x)$ of the buckling operator $\Delta^2-\tau\Delta$, which can be determined uniquely by measuring the Dirichlet-to-Neumann map on subsets of the boundary. We follow the general strategy of \cite{KSU07} using a richer set of solutions to the buckling problem with the Navier boundary conditions.