Counting Singular Matrices with Primitive Row Vectors

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 $T\to \mathrm{\infty }$, 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.