If not mistaken, vector-vector ($R^n$ -> $R^n$) Jacobian can be characterized as $J(X \to Y) = |\det(\frac{\partial x_i}{\partial y_i})|$. Inside the det() is the Jacobian matrix.
Can this be extended to matrix-matrix mapping? Let's say $Y = AX$, where $Y, A$ and $X$ are all of $n\times n$ size. What would be $J(X \to Y)$?