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So I need to prove that: $$\det\left[A^{T}B-B^{T}A\right]=\det[A+B]\cdot\det\left[A-B\right]$$ where $A$, $B$ are two orthogonal matrices, but it seems I'm missing something.

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Hint: Replace $\det(A+B)$ on the right by $\det(A^T+B^T)$ (I trust you understand why that is allowed). Now use the product formula for determinants.

You'll get a problem with signs, but notice that $A^TB-B^TA$ is skew symmetric!

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  • $\begingroup$ Thank you for the answer! Yes I'm aware of the above property but it didn't come to me at that moment, because I was trying to work on the left side of the equality. $\endgroup$ – user209217 Jun 26 '15 at 10:18
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    $\begingroup$ For the signs, just start with $\det[A+B]\det[A-B]=\det[A^T-B^T]\det[A+B]$ instead. $\endgroup$ – Henning Makholm Jun 26 '15 at 11:34
  • $\begingroup$ @HenningMakholm Good point. $\endgroup$ – Harald Hanche-Olsen Jun 26 '15 at 13:00

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