# Inverse of a block 2x2 matrix

How to solve this type of problem: We've got a block 2x2 matrix : $$A=\begin{bmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\\\end{bmatrix}$$ If matrices $A$ and $A_{22}$ are invertible, show that a matrix $B = A_{11} - A_{12}A_{22}^{-1}A_{21}$is also invertible.

• – Shaun Jun 23 '14 at 13:39

Observe that for a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ the claim follows from $(ad - bc)/d = a - bd^{-1}c$.