How to determine if random variables are distributed according to a multivariate normal distribution?

Suppose $(x_1, x_2, x_3)\sim N(\mu, \Sigma)$ where $\mu\in\mathbb{R}^3$ and $\Sigma$ is a $3\times 3$ covariance matrix, are the variables $A = x_1 + x_2$ and $B = x_2 + x_3$ necessarily distributed according to a multivariate normal distribution? That is, do there exist $\nu\in\mathbb{R}^2$ and a $2\times 2$ covariance matrix $M$ such that $(A, B) \sim N(\nu, M)$

Yes and here is how to see it. Write $A$ and $B$ as
$$\begin{bmatrix} A \\B \end{bmatrix}=\begin{bmatrix} 1 & 1 & 0 \\0 & 1 &1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$$