Let $M \in M_n(\mathbb{Z})$ with nonzero determinant. Show that there exists a matrix $A$ in $\mathrm{GL}_n(\mathbb{Z})$ such that $AM$ is lower triangular and, denoting $AM = (c_{ij})$, we have the inequality $0 \leqslant c_{ij} < c_{jj}$ for all $i, j \in \{1, \ldots, n\}$ such that $j < i$.