What is the determinant of a weighted orthogonal projection (based on the weighted pseudo-inverse)?

Question

What is the determinant of a weighted orthogonal projection (based on the weighted pseudo-inverse)? E.g. I have

$$ J = A \left( A^\intercal W A \right)^{-1} A^\intercal W $$

and would like to know $\det (J)$. Note that $A$ is not square while $W$, $J$, and $A^\intercal W A$ are square.

Answer 1

The kernel of any projection onto a proper subspace is nontrivial. If $A$ is not square, then its columns don’t span the entire ambient space, therefore $J$ is rank-deficient and $\det(J)=0$.

Answer 2

Given that $$ \det A^{-1} = \det(A)^{-1} $$ and $$ \det(AB) = \det(A)\det(B) $$ I would say $$ \det((A^T W A)^{-1} A^T W) = \frac{1}{\det A} $$