Show that matrix $A(XB^{-1})^ = $ to $X = C^T(A^{-1})^TB$

Question

I'd like to ask how we arrive from: $A(XB^{-1})^ = $ to $X = C^T(A^{-1})^TB$. A, B and C are all invertible matrixes. I was already thinking to apply the rule $()^ = ^^^$ like the following:

1. $A(XB^{-1})^ = $
2. $A(B^{-1})^TX^T = $
3. $(A(B^{-1})^TX^T)^T = ^T$
4. $XB^{-1}A^T = ^T$

but now I am stuck and don't know how to arrive at $X = C^T(A^{-1})^TB$. How do I solve this (intuitively)?

Answer 1

\begin{alignat*}{3} & \left[A(XB^{-1})^T\right]^T &= &C^T \\ \iff & \big[(XB^{-1})^T\big]^TA^T &= & C^T \\ \iff & XB^{-1}A^T & =& C^T \\ \iff & XB^{-1} &=& C^T(A^T)^{-1}\\ \iff & X &=& C^T(A^T)^{-1}B \end{alignat*}

Using $(A^T)^{-1} = (A^{-1})^T$

$$X = C^T(A^{-1})^TB$$