  We give one more proof of the fact that symplectic matrices over real and
complex fields have determinant one. While this has already been proved many
times, there has been lasting interest in finding an elementary proof. Our
result is restricted to the real and complex case due to its reliance on
field-dependent spectral theory, however in this setting we obtain a proof
which is more elementary in the sense that it is direct and requires only
well-known facts. Finally, an explicit formula for the determinant of conjugate
symplectic matrices in terms of its square subblocks is given.
