Transpose Properties and Operations

Question

This question is rather simple but I'm not quite sure about the answer.

if $(AB)^T= B^TA^T$ then what $(ABC)^T$ equals to?

I ask this in order to solve the following

$(BA^TC+BC)^T (\frac{1}{6}C^TAB^T)^{-1}$ if $A,B, C$ are invertible.

Answer 1

You have

$$ (ABC)^T = ((AB)C)^T = C^T(AB)^T = C^T(B^T A^T) = C^T B^T A^T. $$

Hence, $(BA^TC)^T = C^T (A^T)^T B^T = C^T A B^T$ so

$$(BA^TC)^T \left( \frac{1}{6} C^TAB^T \right)^{-1} = (C^TAB^T) 6 (C^T A B^T)^{-1} = 6I. $$

In addition,

$$ \left( \frac{1}{6} C^T A B^T \right)^{-1} = 6 (B^T)^{-1} A^{-1} (C^T)^{-1} $$

so

$$ (BC)^T \left( \frac{1}{6} C^T A B^T \right)^{-1} = C^T B^T 6 (B^T)^{-1} A^{-1} (C^T)^{-1} = 6 C^T A^{-1} (C^T)^{-1}.$$