  We prove an effective version of a result due to Einsiedler, Mozes, Shah and
Shapira on the asymptotic distribution of primitive rational points on
expanding closed horospheres in the space of lattices. Key ingredients of our
proof include recent bounds on matrix Kloosterman sums due to Erd\'elyi and
T\'oth, results by Clozel, Oh and Ullmo on the effective equidistribution of
Hecke points, and Rogers' integration formula in the geometry of numbers. As an
application of the main theorem, we also obtain a result on the limit
distribution of the number of small solutions of a random system of linear
congruences to a large modulus. Furthermore, as a by-product of our proofs, we
obtain a sharp bound on the number of nonsquare matrices over a finite field
$\mathbb{F}_p$ with small entries and of a given size and rank.
