Determinant of correlation matrix

Last week I needed to know how big determinants of correlation matrices could get¹, and I couldn’t immediately find a proof on Google, so I worked one out.

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Theorem: The determinant of a correlation matrix is at most 1

Proof: Let $p$ be the dimension of the matrix. The trace of the matrix is $p$, and that’s the sum of the eigenvalues, so the arithmetic mean of the eigenvalues is 1. By the arithmetic:geometric mean inequality, the geometric mean is at most 1 (and strictly less if the eigenvalues aren’t all equal). The determinant is the $p$th power of the geometric mean of the eigenvalues, and we are done. ◼️