= Solution
The given $J$ satisfies $J^2=-I_{2m}$ and is invertible. Taking <determinants> in the defining identity of a <symplectic matrix> gives
$$
\det(M^T)\det J\det M=\det J.
$$
Since $\det(M^T)=\det M$ and $\det J\ne0$, cancellation yields $(\det M)^2=1$. Hence
$$
\boxed{\det M=1\text{ or }-1.}
$$
In particular every <symplectic matrix> is invertible. This argument establishes the requested alternatives; the later parts explicitly supply the determinant-one hypothesis that they use.
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