Lemma

Let $\mathbf{R}$ be a positive semidefinite matrix with diagonal elements $r_{ii}, i=1,\dots,k$. Then $$\det \mathbf{R}\le \prod_{i=1}^{k}r_{ii}$$If $\mathbf{R}$ is positive definite, then equality holds if and only if $\mathbf{R}$ is diagonal.