  Let $T$ be an $n\times n$ random matrix, such that each diagonal entry
$T_{i,i}$ is a continuous random variable, independent from all the other
entries of $T$. Then for every $n\times n$ matrix $A$ and every $t\ge0$ $$
\p\Big[|\det(A+T)|^{1/n}\le t\Big]\le2bnt, $$ where $b>0$ is a uniform upper
bound on the densities of $T_{i,i}$.
