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 […]