# Proving two Gaussian random variables are independent given the third: a necessary and sufficient condition for inverse of covariance matrix

In my probability class I was given this problem that truly has me stumped:

Let $X=(X_1,X_2,X_3)$ be a Gaussian random vector with mean vector zeros, with the 3x3 co variance matrix:

$\Sigma$

which is invertible so a joint density function exists. Now we define the following: $p_{13} = {{\Sigma}^{-1}}_{13} = \frac{1}{|\Sigma|} \begin{bmatrix} \Sigma_{12} & \Sigma_{13} \\ \Sigma_{22} & \Sigma_{23} \end{bmatrix}$

that is the {1,3} entry of the inverse of the covariance matrix $\Sigma$. We are to prove for the general 3x3 covariance matrix case that given $X_2$ the random variables $X_1,X_3$ are statistically independent if and only if $p_{13}=0$.

As a hint we are advised to look at the conditional expectation of $X_1$ given $X_2,X_3$ is given by $\mu = \Sigma_{12} {\Sigma_{22}}^{-1} \left( \begin{array}{c} x_2 \\ x_3 \\ \end{array} \right)$

where we define $\Sigma_{12} = \begin{bmatrix} E\{X_1X_2\} & E\{X_1X_3\} \end{bmatrix}$

and we define $\Sigma_{22} = \begin{bmatrix} E\{{X_2}^2\} & E\{X_2X_3\} \\ E\{X_3X_2\} & E\{{X_3}^2\} \end{bmatrix}$

I have no clue as to how to use this clue to show the necessary and sufficient condition for the conditional independence we are given, so I would appreciate anyone giving me help on this here as this clue seems quite vague and unapproachable to me. I thank all kind helpers.