I am having a difficult time understanding how to build a covariance matrix for a multivariate normal.
Let $X=[X_1,X_2,X_3]$ where $E[X_j]=\mu_j, \text{Var}[x_j]=\sigma^2$. $X$ has the multivariate normal distribution,
$$f(x)=\frac{1}{\sqrt{2\pi}^p|\Sigma|}e^{-(x-\mu)'\Sigma^{-1}(x-\mu)}$$
I am given sample values for $\bar{x}_j$ and $S_j^2$, as well as sample size $n_j$. I am ultimately interested in finding a 95% elliptical confidence set for the following equations:
$$z_1=\mu_1+\mu_2-2\mu_3$$ $$z_2=\mu_1-\mu_2$$
I know that I can use the Hotelling $T^2$ to create a rejection region where $n(\bar{x}-\mu)'S^{-1}(\bar{x}-\mu)\leq T_{p,n-1}^2$ for the appropriate $T^2$ value, where S is the variance-covariance matrix approximated with the sample variances. I assume it will be a 3x3 matrix with the individual sample variances along the diagonal, but I am unsure of how to find the covariances off the diagonal.
$$S= \begin{bmatrix} S_1^2 & \sigma_{12} & \sigma_{13}\\ \sigma_{21} & S_2^2 & \sigma_{23}\\ \sigma_{31} & \sigma_{32} & S_3^2 \end{bmatrix} $$
Thank you