Magnetic induction tomography (MIT) attempts to image the electrical and magnetic characteristics of a target using impedance measurement data from pairs of excitation and detection coils. This inverse eddy current problem is nonlinear and also severely ill posed so regularization is required for a stable solution. A regularized Gauss-Newton algorithm has been implemented as a nonlinear, iterative inverse solver. In this algorithm, one needs to solve the forward problem and recalculate the Jacobian matrix for each iteration. The forward problem has been solved using an edge based finite element method for magnetic vector potential A and electrical scalar potential V, a so called A, A - V formulation. A theoretical study of the general inverse eddy current problem and a derivation, paying special attention to the boundary conditions, of an adjoint field formula for the Jacobian is given. This efficient formula calculates the change in measured induced voltage due to a small perturbation of the conductivity in a region. This has the advantage that it involves only the inner product of the electric fields when two different coils are excited, and these are convenient computationally. This paper also shows that the sensitivity maps change significantly when the conductivity distribution changes, demonstrating the necessity for a nonlinear reconstruction algorithm. The performance of the inverse solver has been examined and results presented from simulated data with added noise.