I need help with some questions regarding the normal random vector. Suppose that I have a random vector that follows a multivariate normal distribution
$$ \boldsymbol{X} = \begin{bmatrix} X_1 \\ X_2 \\ X_3 \end{bmatrix} \sim \mathcal{N}_3 \left( \begin{bmatrix} \mu_1 \\ \mu_2 \\ \mu_3 \end{bmatrix}, \begin{bmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{bmatrix} \right) $$
Now, for some reason, I can observe $X_1 = x_1$. I know that the sub-vector $\begin{bmatrix} X_2 \\ X_3 \end{bmatrix} \sim \mathcal{N}_2 \left( \begin{bmatrix} \mu_2 \\ \mu_3 \end{bmatrix}, \begin{bmatrix} \sigma_{22} & \sigma_{23} \\ \sigma_{32} & \sigma_{33} \end{bmatrix} \right)$, but I wonder if it is the case that
$$ \boldsymbol{X} \ \mid X_1 = x_1 \quad \text{or} \quad \begin{bmatrix} x_1 \\ X_2 \\ X_3 \end{bmatrix} \sim \mathcal{N}_3 \left( \begin{bmatrix} x_1 \\ \mu_2 \\ \mu_3 \end{bmatrix}, \begin{bmatrix} 0 & 0 & 0 \\ 0 & \sigma_{22} & \sigma_{23} \\ 0 & \sigma_{32} & \sigma_{33} \end{bmatrix} \right) \quad ? $$
That would be my first question. Another question would be, with the same normal random vector $\boldsymbol{X}$, I know that any linear combination of the components of $\boldsymbol{X}$ is also normally distributed. Specifically,
$$ a_1 X_1 + a_2 X_2 + a_3 X_3 \sim \mathcal{N}_1 \left( \begin{bmatrix} a_1 & a_2 & a_3 \end{bmatrix} \begin{bmatrix} \mu_1 \\ \mu_2 \\ \mu_3 \end{bmatrix}, \begin{bmatrix} a_1 & a_2 & a_3 \end{bmatrix} \begin{bmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{bmatrix} \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} \right) $$
Now, again, I can observe $X_1 = x_1$, and I wonder what the distribution $\boldsymbol{X} \ \mid X_1 = x_1 \quad \text{or} \quad \begin{bmatrix} x_1 \\ X_2 \\ X_3 \end{bmatrix}$ is. I think this is related to my first question. Please give me some insights if you can. Thank you so much.
