  We prove that the defining equations of the Fermat curves of prime degree
cannot be written as the determinant of symmetric matrices with entries in
linear forms in three variables with rational coefficients. In the proof, we
use a relation between symmetric matrices with entries in linear forms and
non-effective theta characteristics on smooth plane curves. We also use some
results of Gross-Rohrlich on the rational torsion points on the Jacobian
varieties of the Fermat curves of prime degree.
