Assume the matrix product term $AXBYC$ where each one is an $n\times n$ matrix. $A, B, C$ are constant, but $X, Y$ are variable and are not independent from one another, hence their covariance is not zero. We know that to calculate the expected value of two random matrices $V, W$ where there are dependent, we have:
$$E[VW]=E[V]E[W]+cov[V,W]$$
My question is how we would get the expectation in this case. I am not sure about the $cov$ part mainly. How would the constants play a part? Can we write $cov(AXB, YC)$ or $cov(AX, BYC)$? Are these different and if so how would we break it down?
$$E[AXBYC]=AE[X]BE[Y]C + ?$$
You cannot do better than observing that the entry $(i,\ell)$ of $E(XBY)$ is $$\sum_{jk}b_{jk}E(X_{ij}Y_{k\ell}),$$