  Let $q=p^n$ be an odd prime power and let $\mathbb{F}_q$ be the finite field
of $q$ elements. Let $\widehat{\mathbb{F}_q^{\times}}$ be the group of all
multiplicative characters of $\mathbb{F}_q$ and let $\chi$ be a generator of
$\widehat{\mathbb{F}_q^{\times}}$. In this paper, we investigate arithmetic
properties of certain cyclotomic matrices involving nonzero squares over
$\mathbb{F}_q$. For example, let $s_1,s_2,\cdots,s_{(q-1)/2}$ be the nonzero
squares over $\mathbb{F}_q$. For any integer $1\le r\le q-2$, define the matrix
  $$B_q(r):=\left[\chi^r(s_i+s_j)+\chi^r(s_i-s_j)\right]_{1\le i,j\le
(q-1)/2}.$$
  We prove that if $q\equiv 3\pmod 4$, then
  $$\det (B_q(r))=\prod_{0\le k\le (q-3)/2}J_q(\chi^r,\chi^{2k})=
  \begin{cases}
  (-1)^{\frac{q-3}{4}}{\bf i}^nG_q(\chi^r)^{\frac{q-1}{2}}/\sqrt{q} &
\mbox{if}\ r\equiv 1\pmod 2,
  G_q(\chi^r)^{\frac{q-1}{2}}/q & \mbox{if}\ r\equiv 0\pmod 2.
  \end{cases},$$
  where $J_q(\chi^r,\chi^{2k})$ and $G_q(\chi^r)$ are the Jacobi sum and the
Gauss sum over $\mathbb{F}_q$ respectively.
