1. ## Multivariate random variable

Let $X$ and $Y$ be random variables and $a,b,c,d \in \mathbb{R}$. When is
$Z=\left[ \begin{matrix}a & b \\ c & d \end{matrix} \right]\left[ \begin{matrix}X \\ Y\end{matrix} \right]$
a multivariate random variable?

1. when $aX+bY\neq cX+dY$
2. always
3. when $ad-bc\neq 0$
4. when $aX+bY=cX+dY$
5. never
6. when $ad-bc=1$
Why is 2nd answer the right one? Just because $Z=\left[ \begin{matrix}aX+bY \\ cX+dY \end{matrix} \right]$ is always a "well defined" (I'm making the terminology up) and, after all, that is the definition of a multivariate random variable (Wikipedia doesn't give one, though)?

2. Because aX+bY and cX+dY are random variables and that makes Z being a multivariate random variable (a vector which components are random variables).
There's nothing else to add o: