If $(X,Y)$ is a Gaussian random vector, is also $(X,Y,1)$ a Gaussian random vector?

Question

if $(X,Y)$ is a Gaussian random vector, can we say that $(X,Y,1)$ is also a Gaussian random vector?

I know that in order to prove this I need to find:

$ \begin{pmatrix}X\\ Y \\ 1 \end{pmatrix} = \underbrace{\begin{pmatrix}a_{11}&a_{12} \\ a_{21}&a_{22} \\ a_{31}&a_{33} \end{pmatrix} }_{A} \begin{pmatrix}X\\ Y \\ \end{pmatrix} $

but the $A$ that I found includes: $\frac{1}{2} X^{-1}$, $\frac{1}{2}Y^{-1}$ in its arguments, and thats not a deterministic matrix.

what I'm missing here?

Answer 1

If we consider any real number as a Gaussian random variable with mean equal to itself and variance equal to $0$, then yes.