Identities with $C$: $MCM^TC=I\det M$, $M+CM^TC=I\operatorname{tr}M$, and $\operatorname{tr}V=0\iff (VC)^T=VC$

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Identities with $C$: $MCM^TC=I\det M$, $M+CM^TC=I\operatorname{tr}M$, and $\operatorname{tr}V=0\iff (VC)^T=VC$ · Prove2Me