Suppose you have vectors X and Y with covariance matrix $V = \left( \begin{array}{cc} A & B \\ B^T & C \end{array} \right)$. This Wikipedia article says that $Var(X | Y) = A - BC^{-1}B^T$, the Schur Complement of C in V.
Furthermore, conditional (in)dependence between X and Y can be found in the inverse of the covariance matrix $V^{-1}$. Earlier in the Wiki page they show that the inverse of a matrix is a function of the Schur Complement.
I'm trying to piece these two pieces together. First, how do we show $Var(X | Y) = A - BC^{-1}B^T$, and second: if $V^{-1} = 0$, how do we show that the Schur complement is necessarily 0, and hence $Var(X | Y) = 0$?