Conditional Multivariate Normal (beyond the bivariate case)

Question

so I need to obtain the conditional distribution of a multivariate normal. However, I can only find it for the bivariate case:

$$(x_1|x_2=a) \sim N(\bar{\mu}, \bar{\Sigma})$$ $$\bar{\mu}= \mu_1 + \Sigma_{12}\Sigma_{22}^{-1}(a-\mu_2)$$ $$\bar{\Sigma}= \Sigma_{11}-\Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21}$$

Here is the example given in more depth.

However, how would these formulas look like if I would have 3 or 4 variables instead of 2?

Edit: I would like the conditional for $(x_1| x_2=a,x_3=b)$ given I have mulitvariate distribution with 3 variables.

Answer 1

so the question was already answered and I just had issues with my Linear Algebra.

Here is the link:

https://stats.stackexchange.com/questions/239317/find-conditional-expectation-from-a-3-dimensional-random-vector?rq=1