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Rearrange $M^{T}JM=J$ asThus $M^{-1}$ is similar to $M^T$. Similar matrices have the same characteristic polynomial by part (a), and transposition also preserves it:Therefore $\boxed{{\chi }_{M^{-1}}={\chi }_M}$. This is the underlying algebraic source of the reciprocal spectrum of a symplectic matrix.