  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.
