2. (15 points) Let $X_1, \dots, X_n$ be jointly normally distributed with mean $\vec{\mu} = \vec{0}$ and covariance matrix $\Sigma = A^{-1}$ ($A$ is called the precision matrix). That is, $X_1, \dots, X_n$ have joint density
$f(x_1, \dots, x_n) = \frac{1}{(2\pi |\Sigma|)^{n/2}} \exp \left( -\frac{1}{2} \sum_{i,j=1}^n x_i A_{ij} x_j \right)$
Show the conditional distribution $f(x_1 | x_2, \dots, x_n)$ is normal and find its mean and variance.
Hint: do not do unnecessary work, including unnecessary integrals. The precision matrix is necessarily symmetric: $A_{ij} = A_{ji}$
