In this article we investigate the relations between the Gorenstein projective dimensions of A-modules and their socles for n-minimal Auslander-Gorenstein algebras A.First we give a description of projective-injective A-modules in terms of their socles.Then we prove that a A-module N has Gorenstein projective dimension at most n if and only if its socle has Gorenstein projective dimension at most n if and only if N is cogenerated by a projective A-module.Furthermore,we show that n-minimal Auslander-Gorenstein al-gebras can be characterised by the relations between the Gorenstein projective dimensions of modules and their socles.