Counting singular matrices with primitive row vectors

We solve an asymptotic problem in the geometry of numbers. where we count the number of singular it n x 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 -->infinity. the number is asymptotic to (n-1)u(n)/zeta(n)zeta(n-1)(n) Tn2-n log (T) for n greater than or equal to 3. The 3-dimensional case is the most problematic and we need to invoke an equidistribution theorem due to W M. Schmidt.