Let $X\in M_n(\Bbb R)$ be a random matrix with iid elements following a continuous distribution.

What are the necessary and sufficient conditions for $$\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$$ to hold? Is there still a geometric interpretation of this inequality?

Examples where the inequality is true:

• $X_{i,j}$ is exponentially distributed, as LHS/RHS equals $(n+1)!/(n^2+1)$

• $X_{i,j}$ is normally distributed with zero mean and variance at least $1/(n!)^{1/n}$, as LHS/RHS equals $\sigma^{2n}n!$

Let $X\in M_n(\Bbb R)$ be a random matrix with iid elements following a continuous distribution.

What are the necessary and sufficient conditions for $$\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$$ to hold? Is there still a geometric interpretation of this inequality?

Examples where the inequality is true:

• $X_{i,j}$ is exponentially distributed, as LHS/RHS equals $(n+1)!/(n^2+1)$

• $X_{i,j}$ is normally distributed with zero mean, as LHS/RHS equals $n!$