# Prove that $\det\left[A^{T}B-B^{T}A\right]=\det[A+B]\cdot\det\left[A-B\right]$

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.

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!
• For the signs, just start with $\det[A+B]\det[A-B]=\det[A^T-B^T]\det[A+B]$ instead. – Henning Makholm Jun 26 '15 at 11:34