  In this paper, by using Cauchy-Binet formula and charcter sums over finite fields, we evaluate determinants of two cyclotomic matrices involving Jacobi sums $J(\chi^i,\chi^j)$. For example, we show that $$\det[J(\chi^i,\chi^j)]_{1\le i,j\le p-2}=(p-1)^{p-3},$$ and that $$\det [J(\chi^{2i},\chi^{2j})]_{1\le i,j\le (p-3)/2} =\frac{p+(-1)^{\frac{p+1}{2}}}{4} \left(\frac{p-1}{2}\right)^{\frac{p-5}{2}}(-1)^{\frac{p+1}{2}},$$ where $p$ is an odd prime and $\chi$ is a generator of the group of all multiplicative characters of $\mathbb{F}_p$. 
On Determinants of cyclotomic matrices involving Jacobi sums