I have a random vector $\mathbf{X}$ consisting of three random variables, $\mathbf{X} = [X_1 X_2 X_3]^T$. I am given a partial covariance matrix C for $\mathbf{X}$, which is $$ C=\begin{bmatrix} 6 & 0 & ? \\ ? & ? & ? \\ 0 & 1 & 4 \end{bmatrix} $$ where the $?$ entry implies we do not have a value. There are three parts I must compute. I have given my approaches below. Are they correct?
a) What are the values of the $?$ entries?
Since $\operatorname{cov}(X_a, X_b) = \operatorname{cov}(X_b, X_a)$, we know right away the off diagonal terms: $$C= \begin{bmatrix} 6 & 0 & 0 \\ 0 & ? & 1 \\ 0 & 1 & 4 \end{bmatrix} $$ I do not believe we can compute $\operatorname{cov}(X_2,X_2)=\operatorname{var}[X_2]$ since $$\operatorname{cov}(X_1,X_2) = E[X_1\cdot X_2] - E[X_1]E[X_2],\quad \operatorname{var}[X_2] = E[X_2^2]-E^2[X^2]$$ (inside outside rule).
I know that covariance matrices are positive semidefinite but I do not see how I can exploit that fact to determine the center term.
b) Are any of the random variables correlated?
Yes, $X_3$ and $X_2$ are correlated because $\operatorname{cov}(X_2,X_3)>0$, otherwise they would be orthogonal (and thus uncorrelated).
c) What is $\operatorname{var}[X_3]$?
$\operatorname{var}[X_3] = 4$ based on the definition of a covariance matrix and $C_{3,3}=4$.