The random vector $X$ is normally distributed with $X \sim \mathcal{N}(\mu,\Omega)$. $\mu $ is a column vector with $(\mu_1, \mu_2, \mu_3)$.
$\Omega$ is $3\times 3$ matrix. $A$ is a $3\times 3$ matrix and $B$ is $2\times 3$ matrix.
Find the distribution of a random vector $W$ (2x1 ) where $W$ =
\begin{bmatrix} A\vec X \\ B\vec X \\ \end{bmatrix}
Link to original (simpler) question.
We can write:$$\left[\begin{array}{c} A\vec{X}\\ B\vec{X} \end{array}\right]=\left[\begin{array}{c} A\\ B \end{array}\right]\vec{X}$$ and characteristic for normal distribution is that this is enough to conclude that the vector again has normal distribution.
This with expectation: $$\mathbb{E}\left[\begin{array}{c} A\\ B \end{array}\right]\vec{X}=\left[\begin{array}{c} A\\ B \end{array}\right]\mathbb{E}\vec{X}=\left[\begin{array}{c} A\\ B \end{array}\right]\vec{\mu}=\left[\begin{array}{c} A\vec{\mu}\\ B\vec{\mu} \end{array}\right]$$
Its covariance matrix is:
$$\left[\begin{array}{c} A\\ B \end{array}\right]\Omega\left[\begin{array}{c} A\\ B \end{array}\right]^{T}=\left[\begin{array}{c} A\\ B \end{array}\right]\Omega\left[\begin{array}{cc} A^{T} & B^{T}\end{array}\right]=\left[\begin{array}{cc} A\Omega A^{T} & A\Omega B^{T}\\ B\Omega A^{T} & B\Omega B^{T} \end{array}\right]$$
The normal distribution is completely determined by expectation and covariance.
