A conjecture of Zhi-Wei Sun on matrices concerning multiplicative subgroups of finite fields

Motivated by the recent work of Zhi-Wei Sun on determinants involving the Legendre symbol, in this paper, we study some matrices concerning subgroups of finite fields. For example, let $q\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$ be an odd prime power and let $\phi$ be the unique quadratic multiplicative character of the finite field ${\mathbb{F}}_q$. If set $\{s_1,\cdots ,s_{(q-1)\mathrm{/}2}\}=\{x^2:x\in {\mathbb{F}}_q\setminus \{0\}\}$, then we prove that $$\det {[t+\phi (s_i+s_j)+\phi (s_i-s_j)]}_{1\le i,j\le (q-1)\mathrm{/}2}=(\frac{q-1}{2}t-1)q^{\frac{q-3}{4}}\mathrm{.}$$ This confirms a conjecture of Zhi-Wei Sun.