Counting $2 \times 2$ integer matrices with a given determinant

Given positive integers $h, N$ satisfying $1 \leqslant h \leqslant 2N^2$, we define $T(h,N)$ to be the number of $2\times 2$ integer matrices with determinant equal to $h$ whose entries lie in $[-N,N]$. Our main result states that for any $\varepsilon >0$, one has $$T(h,N) = \frac{16}{\zeta(2)} N^2 \bigg( \sum_{d |h} \frac{1}{d} \bigg) + O_{\varepsilon}(N^{\varepsilon} (N+ h)).$$ This quantitatively improves upon recent work of Afifurrahman and Ganguly--Guria, and delivers square-root cancellation estimates when $h \leq N$. We further show that when $h$ is large, the error term is of approximately the correct order.