  Motivated by the works of L. Carlitz, R. Chapman and Z.-W. Sun on cyclotomic
matrices, in this paper, we investigate certain cyclotomic matrices concerning
the Jacobi sums over finite fields, which can be viewed as finite field
analogues of certain matrices involving the Beta function. For example, let
$q>1$ be a prime power and let $\chi$ be a generator of the group of all
multiplicative characters of $\mathbb{F}_q$. Then we prove that
$$\det\left[J_q(\chi^i,\chi^j)\right]_{1\le i,j\le q-2}=(q-1)^{q-3},$$ where
$J_q(\chi^i,\chi^j)$ is the Jacobi sum over $\mathbb{F}_q$. This is a finite
analogue of $$\det [B(i,j)]_{1\le i,j\le
n}=(-1)^{\frac{n(n-1)}{2}}\prod_{r=0}^{n-1}\frac{(r!)^3}{(n+r)!},$$ where $B$
is the Beta function. Also, if $q=p\ge5$ is an odd prime, then we show that
$$\det \left[J_p(\chi^{2i},\chi^{2j})\right]_{1\le i,j\le
(p-3)/2}=\frac{1+(-1)^{\frac{p+1}{2}}p}{4}\left(\frac{p-1}{2}\right)^{\frac{p-5}{2}}.$$
