MGF of a Multivariate Distribution This question appeared on one of the challenge questions without solutions on my coursenotes. I couldn't wrap my head around multivariate vectors so any help would be much appreciated. 
Let $Y=(Y_1,Y_2,Y_3)'$ be a $3\times 1$ vector of r.v. having a multivariate distribution ($Y\sim MVN(\mu,\sigma)$). Then the MGF of $Y$ is: 
$$M(t)=\exp\left({\mu't + \frac{t'\sum t}{2}}\right)$$ for $t=\left(t_1,t_2,t_3 \right)'$ 
Now suppose 
$$\mu=\begin{pmatrix} 5\\3\\6\end{pmatrix} \ \ \ \ \ \ \ \ \ \ \ \ \ \sum=\begin{pmatrix} 1 & 1/2 & 1/2 \\ 1/2 & 1 & 1/2 \\ 1/2 & 1/2 & 1 \end{pmatrix}$$
Find MGF of $W=X_2-X_1$ and hence determine the distribution of $W$.
If $V=X_1+X_2$, find the joint MGF of $V$ and $W$ and hence their joint distribution. 
Thanks for any help!
 A: I wouldn't normally do this by using MGFs.  Here's how I'd do it.  (Later, I may figure out the most felicitous way to do it using MGFs directly; then I post that below this.)
$$X_1-X_2 = [1,-1,0]\begin{bmatrix} X_1 \\ X_2 \\ X_3 \end{bmatrix},$$
so
$$
\operatorname{var}(X_1-X_2) = [1,-1,0]\Sigma\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix}\quad\text{(A $1\times 1$ matrix, thus a scalar.)}
$$
and
$$
\mathbb E(X_1-X_2) = [1,-1,0]\mu.
$$
Next, we have
$$
\begin{bmatrix} X_1-X_2 \\ X_1+X_2 \end{bmatrix} = \begin{bmatrix} 1 & -1 & 0 \\ 1 & 1 & 0 \end{bmatrix}\begin{bmatrix} X_1 \\ X_2 \\ X_3 \end{bmatrix}.
$$
So
$$
\operatorname{var} \begin{bmatrix} X_1-X_2 \\ X_1+X_2 \end{bmatrix} = \begin{bmatrix} 1 & -1 & 0 \\ 1 & 1 & 0 \end{bmatrix}\Sigma\begin{bmatrix} 1 & 1 \\ -1 & 1 \\ 0 & 0 \end{bmatrix}\quad\text{A $2\times2$ matrix.}
$$
and
$$
\mathbb E\begin{bmatrix} X_1-X_2 \\ X_1+X_2 \end{bmatrix} = \begin{bmatrix} 1 & -1 & 0 \\ 1 & 1 & 0 \end{bmatrix} \mu.
$$
Once you've found the expected value and the variance, you can plug them into the formula for the MGF.
