Suppose $σ_0, σ_1$ are positive constants and define random variables $X_1, X_2, . . . X_n$ via $X_i =$ $σ_0 Z_0 + σ_1 Z_i$ where $Z_0, Z_1, . . . , Z_n$ each have variance $1$ and are uncorrelated to each other. $(Z_0, Z_1, . . . , Zn_)$ has the standard Gaussian distribution on $R^(N+1)$
Find the variance-covariance matrix $Σ$ of $X_1, X_2, . . . X_n$, and the variance-covariance matrix $\tilde Σ$ of $X_0, X_1, X_2, . . . X_n$, where $X_0 = σ_ 0Z_0$
I am unsure where to start this. I know that a covariance-variance matrix has inputs of the variance of a variable along the diagonals, and the covariance between the two rows/columns on the other entries, but I'm unsure 1) How to calculate those and 2) How to apply that to this specific example.
Thanks in advance!
Update: I've got $Cov(X_i, X_j)$$=Cov(σ_0 Z_0 + σ_1 Z_i)$ $+Cov(σ_0 Z_0 + σ_1 Z_j$) but I'm not sure on advancing from here.