  Let $\mathbb{F}_q$ be an arbitrary finite field of order $q$. In this
article, we study $\det S$ for certain types of subsets $S$ in the ring
$M_2(\mathbb F_q)$ of $2\times 2$ matrices with entries in $\mathbb F_q$. For
$i\in \mathbb{F}_q$, let $D_i$ be the subset of $M_2(\mathbb F_q)$ defined by
  $ D_i := \{x\in M_2(\mathbb F_q): \det(x)=i\}.$ Then our results can be
stated as follows. First of all, we show that when $E$ and $F$ are subsets of
$D_i$ and $D_j$ for some $i, j\in \mathbb{F}_q^*$, respectively, we have
$$\det(E+F)=\mathbb F_q,$$ whenever $|E||F|\ge {15}^2q^4$, and then provide a
concrete construction to show that our result is sharp. Next, as an application
of the first result, we investigate a distribution of the determinants
generated by the sum set $(E\cap D_i) + (F\cap D_j),$ when $E, F$ are subsets
of the product type, i.e., $U_1\times U_2\subseteq \mathbb F_q^2\times \mathbb
F_q^2$ under the identification $ M_2(\mathbb F_q)=\mathbb F_q^2\times \mathbb
F_q^2$. Lastly, as an extended version of the first result, we prove that if
$E$ is a set in $D_i$ for $i\ne 0$ and $k$ is large enough, then we have
\[\det(2kE):=\det(\underbrace{E + \dots + E}_{2k~terms})\supseteq
\mathbb{F}_q^*,\] whenever the size of $E$ is close to $q^{\frac{3}{2}}$.
Moreover, we show that, in general, the threshold $q^{\frac{3}{2}}$ is best
possible. Our main method is based on the discrete Fourier analysis.
