Math Help - covariance

I have learned that the conditional distribution of X1, given that X2 = x2 is normal and has covariance = ∑11 - ∑12 (∑22 )^-1∑21

However, I realized that I was using the fact without knowing why?

How does covariance become like that? I was trying to figure it out from book and internet sources but I couldn't figure it out.

Can anyone help me?

I'm not sure if there is a question here.
But this pdf has some useful info,
http://www.stat.lsu.edu/faculty/mose...37/mvnprop.pdf
and you need to start with normality to get the marginals and conditional distributions to be normal.

thank you for your reply. I just want to know how to prove the covariance