How to remove the parentheses of the matrix product $( (AB)^T (CDE)^T )^T$

Question

How to remove the parentheses of the matrix product $( (AB)^T (CDE)^T )^T$

I recently started doing a course relating to IT practices and as an extra challenge we were told to solve this equation. It has stumped everyone in the course and after scouring the internet for 2 hours I could not find a possible answer. If anyone can help me and give an explanation why I would be very appreciative.

If the answer is $ABCDE$ can you please just say "give dumb dumb gum gum" because that was one of my first instincts but have second guessed myself

Answer 1

In general we have $(XY)^T=Y^TX^T$and $(X^T)^T=X.$

Thus

$((AB)^T(CDE)^T)^T=((CDE)^T)^T((AB)^T)^T=CDEAB.$

Answer 2

Matrix multiplication interacts with the transpose like this: $$(AB)^T=B^TA^T$$ $$(ABC)^T=C^TB^TA^T$$ The order of matrices is reversed. The transpose is also involutory, i.e. $(A^T)^T=A$. So $$((AB)^T(CDE)^T)^T=(B^TA^TE^TD^TC^T)^T=CDEAB$$