Two equalities expressing the determinant of a matrix in terms of expectations over matrix-vector products

We introduce two equations expressing the inverse determinant of a full rank matrix $mathbf{A} in mathbb{R}^{n times n}$ in terms of expectations over matrix-vector products. The first relationship is $|mathrm{det} (mathbf{A})|^{-1} = mathbb{E}_{mathbf{s} sim mathcal{S}^{n-1}}bigl[, Vert mathbf{As}Vert^{-n} bigr]$, where expectations are over vectors drawn uniformly on the surface of an $n$-dimensional radius one hypersphere. The second relationship is $|mathrm{det}(mathbf{A})|^{-1} = mathbb{E}_{mathbf{x} sim q}[,p(mathbf{Ax}) /, q(mathbf{x})]$, where $p$ and $q$ are smooth distributions, and $q$ has full support.