Magnetic properties of a magnetoactive elastomer (MAE) filled with µm-sized soft-magnetic iron particles have been experimentally studied in the temperature range between 150 K and 310 K. By changing the temperature, the elastic modulus of the elastomer matrix was modified and it was possible to obtain magnetization curves for an invariable arrangement of particles in the sample as well as in the case when the particles were able to change their position within the MAE under the influence of magnetic forces. At low (less than 220 K) temperatures, when the matrix becomes rigid, the magnetization of the MAE does not show a hysteresis behavior and it is characterized by a negative value of the Rayleigh constant. At room temperature, when the polymer matrix is compliant, a magnetic hysteresis exists and exhibits local maxima of the field dependence of the differential magnetic susceptibility. The appearance of these maxima is explained by the elastic resistance of the matrix to the displacement of particles under the action of magnetic forces.