I am reading Franz Hohn's Elementary Matrix Algebra (1973) and having trouble solving the following exercise:
Prove that, if $A$ and $B$ are both of order $n$,
(a) $\det A^{T}B = \det A B^T = \det A^T B^T = \det AB$
(b) $\det A^*B^* = \overline{\det AB}$.
My trouble is that the author has not yet proven the multiplicative property $\det AB = \det A \det B.$ If I could use that property (together with $\det A^T = \det A$ and $(AB)^T = B^T A^T$) then the exercise would be trivial. If I could get the first equality in (a) then I could get the rest of the problem.
While attempting to solve this exercise I ended up just proving the multiplicative property, but I don't think that's what the author intends. Am I missing something simple? Any hint is greatly appreciated.
We can use that $\det(C^T)=\det(C)$ and $\det(AB)=\det(BA)$ as follows: $$ \det(A^TB)=\det(A^TB)^T=\det(B^TA)=\det(AB^T) $$ At some point we need to prove the "classical" properties, like $\det(AB)=\det(BA)$ or even better, $\det(AB)=\det(A)\det(B)$. And then we do not care too much whether the text has it at page $xy$ or not.