Counting nonsingular matrices with primitive row vectors

Abstract

We give an asymptotic expression for the number of nonsingular integer $$n\times n$$ -matrices with primitive row vectors, determinant $$k$$ , and Euclidean matrix norm less than $$T$$ , as $$T\rightarrow \infty $$ . We also investigate the density of matrices with primitive rows in the space of matrices with determinant $$k$$ , and determine its asymptotics for large $$k$$ .