Let $A$ be a $n \times m$ matirx and $B$ a $m \times m$ matrix as they are all real-valued. Then does it hold $ \det ( ( A^{T}A ) ( B^T B ) ) = \det ( (AB)^T (AB) ) $ in general?
Do I prove this by mere use of transposition and the property of determinants? If there is a major trick in the proof of it, could you let me know?