Two equalities expressing the determinant of a matrix in terms of expectations over matrix-vector products
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We introduce two equations expressing the inverse determinant of a full rank matrix 𝐀∈ℝ^n × n in terms of expectations over matrix-vector products. The first relationship is |det (𝐀)|^-1 = 𝔼_𝐬∼𝒮^n-1[ ‖𝐀𝐬‖^-n], where expectations are over vectors drawn uniformly on the surface of an n-dimensional radius one hypersphere. The second relationship is |det(𝐀)|^-1 = 𝔼_𝐱∼ q[ p(𝐀𝐱) / q(𝐱)], where p and q are smooth distributions, and q has full support.
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