Past exam of the algebra 1 and differential equations 1 course of the University of Oxford / 2014 / 2 / b / i / Solution

The given $J$ satisfies $J^2=-I_{2m}$ and is invertible. Taking determinants in the defining identity of a symplectic matrix givesSince $\det (M^T)=\det M$ and $\det J\ne 0$, cancellation yields $(\det M{)}^2=1$. HenceIn particular every symplectic matrix is invertible. This argument establishes the requested alternatives; the later parts explicitly supply the determinant-one hypothesis that they use.