We study the differential identities of the algebra Mk(F) of k×k matrices over a field F of characteristic zero when its full Lie algebra of derivations, L=Der(Mk(F)), acts on it. We determine a set of 2 generators of the ideal of differential identities of Mk(F) for k≥2. Moreover, we obtain the exact values of the corresponding differential codimensions and differential cocharacters. Finally we prove that, unlike the ordinary case, the variety of differential algebras with L-action generated by Mk(F) has almost polynomial growth for all k≥2.