The paper concerns the numerical solution of an inverse parametric problem of identifying the diffusion matrix in an elliptic partial differential equation (PDE) from given noisy measurements. The formulated nonlinear ill-posed problem is reduced by applying the least squares approach to an optimization problem with a regularization parameter. Then a full discretization is performed using the finite element method. \(L^2\)-convergence of the numerical solution to the minimal norm solution of the identification problem as the mesh size and the noise level trend to zero is proved. The regularization parameter is coupled to these parameters in a suitable way. The error analysis shows the first order of convergence with respect to the mesh size. A numerical example for a two dimensional inverse problem is presented.