  We study the diophantine exponent of analytic submanifolds of the space of m
by n real matrices, answering questions of Beresnevich, Kleinbock and Margulis.
We identify a family of algebraic obstructions to the extremality of such a
submanifold, and give a formula for the exponent when the submanifold is
algebraic and defined over the rationals. We then apply these results to the
determination of the diophantine exponent of rational nilpotent Lie groups.
