Suppose we have two gaussian vectors $W$ and $Z$ such that their covariance matrix is
$C_{W,Z}$=$$\begin{bmatrix}
4 & \alpha-1 \\
\alpha-1 & 1
\\\end{bmatrix}
$$
the question is find alpha so that $W$ and $Z$ are independent. Do I just replace $\alpha$ with 1 so that $\operatorname{cov}(W,Z)$=0? Am I allowed to do that? Because I can't think of any other way to solve it but we know that if $\operatorname{cov}(X,Y)=0$ it doesn't necessarily mean that $X$ and $Y$ are independent.