  In this paper, we study certain determinants over finite fields. Let
$\mathbb{F}_q$ be the finite field of $q$ elements and let
$a_1,a_2,\cdots,a_{q-1}$ be all nonzero elements of $\mathbb{F}_q$. Let
$T_q=\left[\frac{1}{a_i^2-a_ia_j+a_j^2}\right]_{1\le i,j\le q-1}$ be a matrix
over $\mathbb{F}_q$. We obtain the explicit value of $\det T_q$. Also, as a
consequence of our result, we confirm a conjecture posed by Zhi-Wei Sun.
