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Suppose $\epsilon$ is a $n\times p$ with independent rows $\epsilon_i\sim N(0, \Sigma)$. $Y$ is a matrix of size $n\times p_1$ and $X$ a matrix of size $n\times p_2$ constructed as $$ Y=XA+\epsilon B\\ X=\epsilon C, $$ where $A,B,C$ are matrices of conformable sizes.

My problem is: what is the distribution of $Y|X$?

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