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The conjugacy classification of isometries in $M_o (n)$ depends on the conjugacy of $T$ and $T^{-1}$ in $M_o (n)$. For an element $T$ in $M (n)$, $T$ and $T^{-1}$ are conjugate in $M (n)$, but they may not be conjugate in $M_o (n)$. 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