  We solve an asymptotic problem in the geometry of numbers, where we count the
number of singular $n\times n$ matrices where row vectors are primitive and of
length at most T. Without the constraint of primitivity, the problem was solved
by Y. Katznelson. We show that as $T \to \infty $, the number is asymptotic to
$ \frac{(n-1)u_n}{\zeta (n) \zeta(n-1)^{n}}T^{n^{2}-n}\log (T)$ for $n \ge 3$.
The 3-dimensional case is the most problematic and we need to invoke an
equidistribution theorem due to W. Schmidt.
