Counting Singular Matrices with Primitive Row Vectors By

Abstract. We solve an asymptotic problem in the geometry of numbers, where we count the number of singular n 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!1, the number is asymptotic to ðn1Þun ðnÞðn1Þn T n2n log ðTÞ for n5 3. The 3-dimensional case is the most problem-atic and we need to invoke an equidistribution theorem due to W. M. Schmidt