Statistics of ranks, determinants and characteristic polynomials of rational matrices

We consider the set of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>m</m:mi> <m:mo>×</m:mo> <m:mi>n</m:mi> </m:mrow> </m:math> {m× n} matrices with rational entries having numerator and denominator of size at most H and obtain various upper bounds on the number of such matrices of a given rank, or with a given determinant, or a given characteristic polynomial. We also consider similar questions for matrices whose entries are Egyptian fractions.