# Normal random vector with some known values [duplicate]

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.