An optimal bound on the number of determinants generated by a subset in $\mathbb{F}_q^d$

In this short note, we prove that for $\mathcal{E} \subset \mathbb{F}_q^d$ with $|\mathcal{E}| \geq q^{d-1} + O(q^2)$ then the set of determinants generated by $\mathcal{E}$ is $\mathbb{F}_q.$ This result is nearly optimal and generalizes the previous results of Vinh (2013) and Iosevich, Rudnev and Zhai (2015).