I am working through Terras' Harmonic Analysis, V2, and am stuck on I believe a notational point. We are asked to show that for $$Y=\begin{pmatrix}V&0\\0&W\end{pmatrix}\begin{bmatrix}I_p&0\\X&I_q\end{bmatrix},$$ then $$J:=\left|\frac{\partial Y}{\partial(V,W,X)}\right|=|W|^p.$$
I assume that $J$ is the determinant of the Jacobian, but I am not sure how to take this, since $V,W,X$ are matrices.
A couple notes: this is excercise 19 in sec. 4.1.3; $A[B]=B^tAB$.
The key is to treat the matrices as vectors. One obtains $$\frac{\partial Y}{\partial(V,W,X)}= \begin{pmatrix} I_{p^2}&\frac{\partial (W[X])}{\partial W}&\frac{\partial (W[X])}{\partial X}\\ 0&\frac{\partial (WX)}{\partial W}&\frac{\partial (WX)}{\partial X}\\ 0&I_{q^2}&0 \end{pmatrix}. $$ Then all that remains is taking $\left|\frac{\partial(WX)}{\partial X}\right|$.