The question is: Assuming that all the following matrices are of the same size and nonsingular, solve $AB(A^{-1})(D^T)(C-1 )= E$ for matrix $D$.
So far I got to $D^T = EC(B^{-1})$, but I do not know if it would be $D=[EC(B^{-1})]^{-T}$, or $D=[EC(B^{-1})]^T$, or even something else?
Think again! Matrices do not commute in general. So what you should get is $$ D^{T} = A B^{-1} A^{-1} C, $$ and then transpose, which inverts the order of the factors.
(I have taken $E$ to be the identity matrix, if it's another generic matrix, then you get $D^{T} = A B^{-1} A^{-1} E C$.)