Let $\hat{B} \sim N(B, (X^TX)^{-1}\sigma), P(X|G=k) \sim N(u_k, \Sigma_k), \hat{B} = (X^TX)^{-1}X^TY$ (assume $\hat{B}$ is an invertible, square matrix) where $Y$ is the indicator response matrix. We want to show that $P(Y|G=k)$ is normally distributed. Where $Y=X\hat{B}$.
$P(Y \le y|G=k) = P(X\hat{B} \le y|G=k) = P(X \le \hat{B}^{-1}y|G=k)$
$P(Y|G=k) = P_x(\hat{B}^{-1}y|G=k)|\hat{B}^{-1}|$
How can one compute the $|\hat{B}^{-1}|$ part?