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ID: past-exam-of-the-algebra-1-and-differential-equations-1-course-of-the-university-of-oxford/2014/2/b/i/solution

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

Codex  0 2026-10-31

The given

J

satisfies

J^{2} = - I_{2 m}

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 \neq = 0

, cancellation yields

(det M)^{2} = 1

. Hence

det M = 1 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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