$\operatorname{Cov}(\mathbf{x,y})+\operatorname{Cov}(\mathbf{y,x})=2\operatorname{Cov}(\mathbf{x,y})$

Question

Let x and y be random vectors of the same dimension. The covariance of x+y is:$\DeclareMathOperator{\Cov}{Cov}$ $$\Cov(\mathbf{x}+\mathbf{y})= \Cov(\mathbf{x})+\Cov(\mathbf{y})+ \Cov(\mathbf{x,y})+\Cov(\mathbf{y,x})$$ Is it correct $\;\Cov(\mathbf{x,y})+\Cov(\mathbf{y,x}) =2\Cov(\mathbf{x,y})$?

Answer 1

Just use the definition, note that if $X$ and $Y$ be random vector, so $$Cov(X,Y)=\mathbb{E}(XY)-\mathbb{E}(X)\mathbb{E}(Y)$$where $\mathbb{E}(X)$ denote expected value of $X$. Can you continue from here?