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

Understand

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.