Linear combination of two dependent multivariate normal random variables Suppose we have two vectors of random variables, both are normal, i.e., $X \sim N(\mu_X, \Sigma_X)$ and $Y \sim N(\mu_Y, \Sigma_Y)$. We are interested in the distribution of their linear combination $Z = A X + B Y + C$, where $A$ and $B$ are matrices, $C$ is a vector. If $X$ and $Y$ are independent, $Z \sim N(A \mu_X + B \mu_Y + C, A \Sigma_X A^T + B \Sigma_Y B^T)$. The question is in the dependent case, assuming that we know the correlation of any pair $(X_i, Y_i)$. Thank you.
Best wishes,
Ivan
 A: Your question does not have a unique answer as currently posed unless you assume that $X$and$Y$ are jointly normally distributed with covariance top right block $\Sigma_{XY}$.  i think you do mean this because you say you have each covariance between X and Y.  In this case we can write $W=(X^T,Y^T)^T$ which is also multivariate normal.  then $Z$ is given in terms of $W$ as:
$$Z=(A,B)W+C$$
Then you use your usual formula for linear combination.  Note that the mean is unchanged but the covariance matrix has two extra terms added $A\Sigma_{XY}B^T+B\Sigma_{XY}^TA^T$
A: In that case, you have to write (with hopefully clear notations)
$$
\left(\begin{matrix}X\\Y \end{matrix}\right) \sim \mathcal{N}\left[ \left(\begin{matrix}\mu_X\\\mu_Y\end{matrix}\right), \Sigma_{X,Y} \right]
$$
(edited: assuming joint normality of $(X,Y)$)
Then
$$
AX+BY=\left(\begin{matrix}A& B \end{matrix}\right)
\left(\begin{matrix}X\\Y \end{matrix}\right)
$$
and
$$
AX+BY+C \sim \mathcal{N}\left[ 
\left(\begin{matrix}A& B \end{matrix}\right)
\left(\begin{matrix}\mu_X\\\mu_Y\end{matrix}\right) + C, 
\left(\begin{matrix}A & B \end{matrix}\right)\Sigma_{X,Y} \left(\begin{matrix}A^T \\ B^T \end{matrix}\right)\right]
$$
i.e.
$$
AX+BY+C \sim \mathcal{N}\left[A\mu_X + B\mu_Y +C,
A\Sigma_{XX}A^T+B\Sigma_{XY}^TA^T+A\Sigma_{XY}B^T+B\Sigma_{YY}B^T
\right]
$$
