Suppose that $A$ is an $m\times m$ matrix, and $S$ a $m\times n$ matrix. Is it possible for the relation $$ \det S^{\dagger}A S= \det A \det S^{\dagger}S$$ to hold also for $n\neq m$?
For instace, if $\det A = 0$ is it true that also $\det S^{\dagger}AS = 0$?
The equality does not hold in general. Consider the following counter-example:
$A=\begin{bmatrix} 1 & 0\\ 0 & x \end{bmatrix}$ and $S=\begin{bmatrix} y \\ 0 \end{bmatrix} .$ Then note that $$ \det (S^{\dagger}AS) = y^2,\quad \det(A)\det(S^{\dagger}S)=xy^2. $$Picking suitable $x$ and $y$ you can see that the equality does not hold. You can also take $x=0$ if you need a non-invertible example.