Multiplication of random matrices with independent entries

Question

Let A and B be two random matrices with zero-mean i.i.d. entries. Then, are the entries of C = A*B independent? From intuition, each entry of C is the dot product of two different independent random vectors. But, I am not quite sure how to formally show whether the entries are/are not independent.

Answer 1

The entries of $AB$ need not be independent.

As an example, let $A,B$ be $2{\,\times\,}2$ matrices with entries chosen independently from a uniform distribution on $\{-1,0,1\}$. For two such random matrices $A,B$, if it happens to be the case that the entries on the main diagonal of $AB$ are both equal to $2$, then all entries of $AB$ must be even. But with no given information, the probability that an entry of $AB$ is equal to $\pm 1$ is positive.