There is a natural left and right invariant Haar measure associated with the matrix groups GL N (R) and SL N (R) due to Siegel. For the associated volume to be finite it is necessary to truncate the groups by imposing a bound on the norm, or in the case of SL N (R), by restricting to a fundamental domain. We compute the asymptotic volumes associated with the Haar measure for GL N (R) and SL N (R) matrices in the case of that the operator norm lies between R 1 and 1/R 2 in the former, and this norm, or alternatively the 2-norm, is bounded by R in the latter. By a result of Duke, Rundnick and Sarnak, such asymptotic formulas in the case of SL N (R) imply an asymptotic counting formula for matrices in SL N (Z). We discuss too the sampling of SL N (R) matrices from the truncated sets. By then using lattice reduction to a fundamental domain, we obtain histograms approximating the probability density functions of the lengths and pairwise angles of shortest length bases vectors in the case N = 2 and 3, or equivalently of shortest linearly independent vectors in the corresponding random lattice. In the case N = 2 these distributions are evaluated explicitly. arXiv:1604.07462v1 [math-ph]