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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?

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    @hardmath Yep, by your note I've solved the question. Thanks!2012-10-12

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$ |A^{T}A||B^{T}B| = |A^{T}A||B^{T}||B| = |B^{T}||A^{T}A||B| $ which equals $ |B^{T}(A^{T}A)B| = |(B^{T}A^{T})(AB)| = |(AB)^{T}AB|. $