Let $(X_1,X_2,X_3)$ be normally distributed with
\begin{align} \mu = \begin{pmatrix} \mu_1 \\ \mu_2 \\ \mu_3 \end{pmatrix} \text{, } \quad\Sigma = \begin{pmatrix} 1 &\rho &0 \\ \rho &1 &\rho \\ 0 &\rho &1 \end{pmatrix} \quad (\rho^2 < 1/2) \end{align}
What is the conditional distribution of $X_2$ given $X_1 = x_1$, $X_3 = x_3$? Under what condition does this distribution coincide with the marginal distribution of $X_2$?
I am sure that this is meant to be solved by the following theorem from the lecture:
If $x \sim \mathcal{N}(\mu,\Sigma)$ with $$ \mu = \begin{pmatrix} \mu_1 \\ \mu_2 \end{pmatrix}, \quad \Sigma = \begin{pmatrix} \Sigma_{11} &\Sigma_{12} \\ \Sigma_{21} & \Sigma_{22} \end{pmatrix}$$ then the conditional distribution of any subset vector $x_1$, given the complement vector $x_2$ is also a multivariate normal distribution given by $$x_1 \vert x_2 \sim \mathcal{N}(\mu_{1 \vert 2}, \Sigma_{1 \vert 2}),$$ where
$$\mu_{1 \vert 2} = \mu_1 + \Sigma_{12}\Sigma_{22}^{-1}(x_2-\mu_2)$$ $$\Sigma_{1 \vert 2} = \Sigma_{11}-\Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21}$$
What I do not get here is how I should write down the $\mu$ and $\Sigma$ so that I can use the theorem from the lecture. Could you please give me a hint?