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Everywhere it is done for bivariate or given a hints. But Not in details for trivariate.

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  • $\begingroup$ The second part of this question is a strong hint: because $Y$ is multivariate Normal, the regression of $Y_1$ on $(Y_2,Y_3)$ is linear and its coefficients are determined by $\Sigma.$ The linear regression determines the conditional distribution. $\endgroup$
    – whuber
    Feb 1, 2019 at 15:54
  • $\begingroup$ I can't understand your answer. Is it possible to solve this problem? @whuber $\endgroup$ Feb 1, 2019 at 16:57

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