Suppose $X\sim N(\mu,V)$ where
$\mu = \begin{pmatrix} 2 \\ 2 \\ 2 \end{pmatrix}$
$V = \begin{pmatrix} 3 & 2 & 1 \\ 2& 4 & 1 \\ 1 & 1 & 2 \end{pmatrix}$
a) Write down the p.d.f of $X$
b) Find the distribution of $AX$ where $A= \begin{pmatrix} 2 & -1 & -1 \\ 0 & 1 & -1 \\ \end{pmatrix}$
c) Find the distribution of $BX$ where $B= \begin{pmatrix} 1 & 1 & 1 \\ 1 & -1 & 0 \\ 0 & 1 & -1 \\ \end{pmatrix}$
d) Find the marginal distributions of $(X_1,X_2)$ and of $X_3$
for a) I have
$ f(x) =$ $ 1 \over(2\pi)^{3/2} |V|^{1/2}$$e^{-1/2(x-\mu)^TV^-1(x-\mu)}$
I don't know if it's right
b) mean: $\begin{pmatrix} 2 & -1 & 1 \\ 0 & 1 & -1 \\ \end{pmatrix}$$\begin{pmatrix} 2\\ 2\\ 2\\ \end{pmatrix}=$$\begin{pmatrix} 0\\ 0\\ \end{pmatrix}$
covariance: $\begin{pmatrix} 2 & -1 & -1 \\ 0 & 1 & -1 \\ \end{pmatrix}$ $\begin{pmatrix} 3 & 2 & 1 \\ 2 & 4 & 1 \\ 1 & 1 & 2 \\ \end{pmatrix}$ $\begin{pmatrix} 2 & 0 \\ -1 & 1 \\ -1 & -1 \\ \end{pmatrix}$$=\begin{pmatrix} 8 & 0 \\ 0 & 4 \\ \end{pmatrix}$
$AX \sim N($$\begin{pmatrix} 0\\ 0\\ \end{pmatrix},$$\begin{pmatrix} 8 & 0 \\ 0 & 4 \\ \end{pmatrix})$
c) mean: $\begin{pmatrix} 1 & 1 & 1 \\ 1 & -1 & 0 \\ 0 & 1 & -1 \\ \end{pmatrix}$$\begin{pmatrix} 2\\ 2\\ 2\\ \end{pmatrix}=$$\begin{pmatrix} 6\\ 0\\ 0\\ \end{pmatrix}$
covariance: $\begin{pmatrix} 1 & 1 & 1 \\ 1 & -1 & 0 \\ 0 & 1 & -1 \\ \end{pmatrix}$ $\begin{pmatrix} 3 & 2 & 1 \\ 2 & 4 & 1 \\ 1 & 1 & 2 \\ \end{pmatrix}$ $\begin{pmatrix} 1 & 1 & 0 \\ 1 & -1 & 1 \\ 1 & 0 & -1 \\ \end{pmatrix}$$=\begin{pmatrix} 17 & -1 & 3 \\ -1 & 3 & -2 \\ 3 & -2 & 4 \\ \end{pmatrix}$
$BX \sim N($$\begin{pmatrix} 6\\ 0\\ 0\\ \end{pmatrix},$$\begin{pmatrix} 17 & -1 & 3 \\ -1 & 3 & -2 \\ 3 & -2 & 4 \\ \end{pmatrix})$
d) $(X_1,X_2) \sim N($$\begin{pmatrix} 2 \\ 2 \\ \end{pmatrix},$$\begin{pmatrix} 3 & 2 \\ 2 & 4 \\ \end{pmatrix})$
$X_3 \sim N(2,2)$
Am I right?