Counting $2\times 2$ matrices with fixed determinant and bounded coefficients

Recent work by M. Afifurrahman established the first asymptotic estimates with error terms for the number of $2\times 2$ matrices with fixed non-zero determinant $n\in\mathbb{N}$, and with coefficients bounded in absolute value by $X$. In this paper we present a new proof of this result, which also gives an improved error term as $X\rightarrow\infty$. Similar to Afifurrahman's result, our error term is uniform in both $n$ and $X$, and our estimates are significant for $X$ as small as $n^{1/2+\delta}$. To complement this, we also demonstrate that the exponent $1/2+\delta$ in this statement cannot be reduced, by establishing a result which gives a different asymptotic main term when $n$ is either a prime or the square of a prime, and when $X=n^{1/2}$.