Relationship between the expected value of the determinant and the determinant of the expected value?

Question

Let $X$ be an $n \times n$ matrix of random variables. Is $E[\det(X)] = \det(E[X])$?

If it doesn't hold in general, under what assumptions on $X$ does it hold?

Answer 1

It is not true in general. Let $X= [A, B ; C , D] \in \mathbb{R}^{2\times 2}$. Then the statement is equivalent with $E[ AD - BC ] = E[A]E[D]- E[B]E[C]$. If $B$ and $C$ are uncorrelated, but $A$ and $D$ are correlated, this is false. @yotam maoz is right though that it does hold in the case when the entries are independent. As suggested by @Lorenzo Najt, if only rather the rows or columns are independent, then the same holds, since when calculating the determinant, one can expand along the rows or the columns, calculating the determinant by multiplying each entry in a row/column by the determinant of the corresponding minorant matrix, which will be comprised of sums/products of uncorrelated terms.