Show that matrix $[A, B; B^T, C]$ is nonsingular

Question

For homework at my uni we have the following problem:

Let $A$, $B$ and $C$ all be $n\times n$ matrices. Suppose $C$ and $A-BC^{-1}B^T$ are both nonsingular. Show that matrix $[A B; B^T C]$ is nonsingular and find its inverse.

Now I know when a matrix is nonsingular or not. But with this problem I don't know where or how to start, I have tried many things but all of them failed. Could someone help me?

Answer 1

What you're looking at is the Schur complement of $C$ in your block matrix $M$. See the wiki page for a formula for the inverse.

The idea here is to apply block-matrix row-reduction.