If $A$ is orthogonal, $(A-A^{-1})^T=A-A^T\neq -(A-A^{-1})=A^{-1}-A$
If $A$ is involutory, do we have an exception? In that case $(A-A^{-1})=0$, which seems trivial.
2026-03-25 17:52:55.1774461175
Is $(A-A^{-1})$ skew-symmetric?
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Your statement that "if $A$ is orthogonal, then $(A - A^{-1})$ is skew-symmmetric" is indeed correct and is still correct when $A$ is additionally symmetric since the zero-matrix "counts" as being skew-symmetric.