Let $\mathbf x$ be a $n\times1$ random variable, $\mathbf s$ be a vector of size $3\times 1$ and $A$, $M_1$, $M_2$ and $M_3$ be $n\times n$ matrices. What is the following expectation with respect to $\mathbf x$
$$ y =E_x \Big[\parallel A\mathbf x+[M_1\mathbf x, M_2\mathbf x, M_3\mathbf x]\mathbf s\parallel ^2\Big] $$
My work: $$y =E_x \Bigg[\bigg( A\mathbf x+\bigg[M_1\mathbf x, M_2\mathbf x, M_3\mathbf x\bigg]\mathbf s\bigg)^T \bigg(A\mathbf x+\bigg[M_1\mathbf x, M_2\mathbf x, M_3\mathbf x\bigg]\mathbf s\bigg)\Bigg] $$
$$=E_x \Bigg[ \mathbf x^TA^TA\mathbf x+ \mathbf x^TA^T\bigg[M_1\mathbf x, M_2\mathbf x, M_3\mathbf x\bigg]\mathbf s+ s^T\bigg[M_1\mathbf x, M_2\mathbf x, M_3\mathbf x\bigg]^TA\mathbf x+ s^T\bigg[M_1\mathbf x, M_2\mathbf x, M_3\mathbf x\bigg]^T\bigg[M_1\mathbf x, M_2\mathbf x, M_3\mathbf x\bigg]\mathbf s \Bigg] $$
$$=E_x \Big[\mathbf x^TA^TA\mathbf x\Big]+ E_x \Bigg[\mathbf x^TA^T\bigg[M_1\mathbf x, M_2\mathbf x, M_3\mathbf x\bigg]\mathbf s\Bigg]+ E_x \Bigg[s^T\bigg[M_1\mathbf x, M_2\mathbf x, M_3\mathbf x\bigg]^TA\mathbf x\Bigg]+ E_x \Bigg[s^T\bigg[M_1\mathbf x, M_2\mathbf x, M_3\mathbf x\bigg]^T\bigg[M_1\mathbf x, M_2\mathbf x, M_3\mathbf x\bigg]\mathbf s \Bigg]$$
Since $\mathbf x^TA^T\bigg[M_1\mathbf x, M_2\mathbf x, M_3\mathbf x\bigg]\mathbf s = s^T\bigg[M_1\mathbf x, M_2\mathbf x, M_3\mathbf x\bigg]^TA\mathbf x$,
$$ y = E_x \Big[\mathbf x^TA^TA\mathbf x\Big]+ 2E_x \Bigg[\mathbf x^TA^T\bigg[M_1\mathbf x, M_2\mathbf x, M_3\mathbf x\bigg]\mathbf s\Bigg]+ E_x \Bigg[s^T\bigg[M_1\mathbf x, M_2\mathbf x, M_3\mathbf x\bigg]^T\bigg[M_1\mathbf x, M_2\mathbf x, M_3\mathbf x\bigg]\mathbf s \Bigg]\tag{1}$$
I know if $\Lambda$ be a symmetric matrix,
$$E_x[\mathbf x^T\Lambda \mathbf x] = \mu_x^T\Lambda \mu_x + tr(\Lambda\Sigma_x )\tag2$$
where $\mu_x = E[\mathbf x]$ and $\Sigma_x = Var(\mathbf x)$.
Using $(2)$, the first term of $(1)$ will be
$$E_x \Big[\mathbf x^TA^TA\mathbf x\Big] = \mu_xA^TA\mu_x+tr(A^TA\Sigma_x)\tag{3}$$ But I have no idea to go on. I don't know if $A^TM_i$ is symmetric or not. Can you help me assuming (i) $A^TM_i$ is symmetric, (ii) $A^TM_i$ is not symmetric? Thanks.