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Vector of multivariate normal distribution

Let $\textbf{X} = (X_1, X_2, X_3)^T$ and $\textbf{Y} = (Y_1, Y_2, Y_3)^T$ be independent vectors with multivariate normal distribution, with means $\mu_X$ and $\mu_Y$ and covariance matrices $\Sigma_X$ and $\Sigma_Y$ with non-zero determinant. Let $A_{2 \times 3}$ and $B_{3 \times 3}$ be lineary independent matrices. Find distribution of $(\textbf{X}^TA^T, \textbf{Y}B^T)^T$.

This is what I’ve done

Let $\textbf{Z} = (\textbf{X}^T, \textbf{Y}^T )^T$. It has multivariate normal distribution with mean $\mu_{Z}$ = $\big(\begin{smallmatrix}
\mu_{X}\\
\mu_{Y}
\end{smallmatrix}\big)$
and covariance matrix $\Sigma_{Z} = \big(\begin{smallmatrix}
\Sigma_{X} & 0\\
0 & \Sigma_Y
\end{smallmatrix}\big)$
.

Let C = $\big(\begin{smallmatrix}
A & 0\\
0 & B
\end{smallmatrix}\big)$
.

Then $(\textbf{X}^TA^T, \textbf{Y}B^T)^T = C\textbf{Z}$ with multivariate normal distribution $N(C\mu_{Z}, C\Sigma_{Z}C^T)$.

Can you tell me if it is good solution or if it isn’t where did I make a mistake?

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