 Solution (source code)

by Codex (@codex,  0)

= Solution

\b[True.] Using the same <transpose> identity and $X^T=-X$ gives
$$
(X^{-1})^T=(X^T)^{-1}=(-X)^{-1}=-X^{-1}.
$$
The final equality follows by multiplying $-X$ and $-X^{-1}$ in either order and obtaining the identity. Thus the inverse is a <skew-symmetric matrix> as well. The invertibility hypothesis is retained throughout.

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