We explore the effects of dissipationless merging on the Fundamental Plane of elliptical galaxies by using a N-body code based on a new, high performance numerical scheme (Dehnen 2002). We investigate the two extreme cases of galaxy growth by equal mass merging and accretion of small stellar systems; in a subset of simulations we also consider the presence of dark matter halos around the merging galaxies. Curiously, we found that the Fundamental Plane is preserved by major merging, while in the accretion scenario its edge–on thickness is only marginally reproduced, with substantial thickening in the case of merging with low angular momentum. We also found that both the Faber-Jackson and Kormendy relations are not reproduced by the simulations, in accordance with the results of a preliminary analysis based on a simple application of the virial theorem. Finally, we discuss the implications of our results for the origin of the MBH-σ0 and Magorrian relations. We found that dissipationless merging is unable to reproduce the MBH-σ0 relation, if the black hole masses add linearly (while the Magorrian relation is nicely reproduced); on the contrary a black hole merging with substantial emission of gravitational waves reproduces the MBH-σ0 relation but fails at reproducing the Magorrian relation. We argue that our results strongly point towards a major role of dissipation in the formation of early–type galaxies and in the growth of their central supermassive black holes, thus supporting the idea of a link between galaxy formation and QSO activity.