Consider two independent random variables $ξ_1$ and $ξ_2$, such that $ξ_1 ∼ N(0,1)$ and $ξ_2 ∼ N(0,2)$. Let $η_1 =(ξ_1+ξ_2, ξ_2)^{T}$, $η_2 =(ξ_1, ξ_1−ξ_2)^{T}$. Find the covariance matrix between $η_1$ and $η_2$.
How would I go about answering this?
\begin{align} cov(\eta_2, \eta_1) &= \mathbb{E} \eta_2^T \eta_1 - \mathbb{E}\eta_2\mathbb{E}\eta_1\\ &=\mathbb{E} \begin{bmatrix}\xi_1^2 + \xi_1\xi_2,&\xi_1\xi_2\\ \xi_1^2 - \xi_2^2,&\xi_1\xi_2-\xi_2^2 \end{bmatrix}\\ &=\begin{bmatrix}1,&0\\-1,&-2 \end{bmatrix} \end{align}