Last week I needed to know how big determinants of correlation matrices could get^{1}, and I couldn’t immediately find a proof on Google, so I worked one out.

**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. ◼️

yes, I know, you probably learned this primary school↩︎