  A rank-$r$ integer matrix $A$ is $\Delta$-modular if the determinant of each
$r \times r$ submatrix has absolute value at most $\Delta$. The class of
$1$-modular, or unimodular, matrices is of fundamental significance in both
integer programming theory and matroid theory. A 1957 result of Heller shows
that the maximum number of nonzero, pairwise non-parallel rows of a rank-$r$
unimodular matrix is ${r + 1 \choose 2}$. We prove that, for each sufficiently
large integer $r$, the maximum number of nonzero, pairwise non-parallel rows of
a rank-$r$ $2$-modular matrix is ${r + 2 \choose 2} - 2$.
