It is a natural question to count matrices
$A$ with integer entries in an expanding box of side length
$\det(A) = r$, a fixed integer; or with the characteristic polynomial of
$A = f$, a fixed integer polynomial; and there are several results in the literature on these problems. Most of the existing results, which use either Ergodic methods or Harmonic Analysis, give asymptotics for the number of such matrices as
$x$ goes to infinity and in the only result we have been able to find that gives a bound on the error term, the bound is not very satisfactory. The aim of this talk will be to present an ongoing joint work with Rachita Guria in which, for the easiest case of
$2 \times 2$ matrices, we have been able to obtain reasonable bounds for the error terms for the above problems by employing elementary Fourier Analysis and results from the theory of Automorphic Forms.