  In this paper we study some determinant inequalities and matrix inequalities
which have a geometrical flavour. We first examine some inequalities which
place work of Macbeath [13] in a more general setting and also relate to recent
work of Gressman [8]. In particular, we establish optimisers for these
determinant inequalities. We then use these inequalities to establish our main
theorem which gives a geometric inequality of matrix type which improves and
extends some inequalities of Christ in [5].
