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?


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.

  • $\begingroup$ I see now that this is the schur complement, but I have no idea how block-matrix row-reduction works nor how to show that this matrix is nonsingular or its inverse. $\endgroup$ – Thom van Essen Dec 8 '16 at 15:20
  • $\begingroup$ Take a look at the article, then, and see if that makes sense to you. $\endgroup$ – Omnomnomnom Dec 8 '16 at 15:22
  • $\begingroup$ After reading that article and reading some stuff about block-matrix inversion I think I know how to solve this. Thanks a lot. $\endgroup$ – Thom van Essen Dec 8 '16 at 15:37
  • $\begingroup$ You're welcome. Once you're confident that you've arrived at a solution, I'd appreciate it if you could accept this answer (by clicking the $\checkmark$). $\endgroup$ – Omnomnomnom Dec 8 '16 at 15:43

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