How well can a random point on a submanifold of $R^n$ be approximated by points with rational coordinates ? In the 90’s Kleinbock and Margulis introduced a new method to answer this question, based on dynamics of diagonal flows on the space of lattices. Roughly speaking they showed that if the submanifold is analytic and not degenerate, i.e. not contained in a proper affine subspace, then a random point on it behaves just as well as a random point in the ambient $R^n$ as far as diophantine approximation is concerned. In this talk I will study the analogous problem when the ambient space is the space of $n x m$ matrices and diophantine approximation is understood with respect to matrix multiplication. In particular I will present a simple algebraic criterion on the submanifold analogous to the above non-degeneracy condition ensuring that random points are well-behaved; this had been a problem for some time. The method actually yields exact diophantine exponents depending on some simple algebraic data regarding the submanifold. As an application we compute the diophantine exponents of random $k$-generated subgroups of nilpotent Lie groups. Joint work with Menny Aka, Lior Rosenzweig and Nicolas de Saxce.