Let $X=(X_1, X_2)$ be a random vector with $X$~$\mathcal{N}_2(\mu, \Sigma)$, $\mu$ is a $2\times1$ matrix and $\Sigma$ is a $2\times2$ matrix. Let $Y=X_1+2X_2$, $Z=(X_1+X_2, X_2)$, find the distribution of Y and Z.
For Y, I have $E[Y]=E[(1 \quad 2)X]=(1 \quad 2)E[X]=(1 \quad 2)\mu=1\mu_1+2\mu_2$ and ${\rm Var}[Y]={\rm Var}[(1 \quad 2)X]=(1 \quad 2)\Sigma(1 \quad 2)^T$, with $Y$~$\mathcal{N}_2(E[Y], {\rm Var}[Y])$
For Z, I have $E[Z]=E[(X_1+X_2, X_2)]=(E[X_1]+E[X_2], E[X_2])=(\mu_1+\mu_2, \mu_2)$.
How can I find ${\rm Var}[Z]$?
Note the distribution of $Y$ is univariate normal.
For the distribution of $Z$, here's a hint: find a $2\times 2$ matrix $A$ so that $Z=AX.$
Now use the fact that affine transforms of normal random vectors are also normal, as you did with the distribution of $Y$.