Paragraph

Suppose that onc can partition the matrix \(A\) as

\begin{equation*} A = \begin{pmatrix} A_{11} \amp A_{12} \\ A_{21} \amp A_{22}, \end{pmatrix} \end{equation*}

where \(A\) and \(A_{11}\) are nonsingular. Let \(S=A_{22} - A_{21}A_{11}^{-1}A_{12}\text{.}\) Show that

\begin{equation*} A^{-1} = \begin{pmatrix} A_11^{-1} + A_{11}^{-1}A_{12}S^{-1}A_{21}A_{11}^-1 \amp -A_{11}^{-1}A_{12}S^{-1}\\ -S^{-1}A_{21}A_{11}^{-1} \amp S^{-1}. \end{pmatrix} \end{equation*} in-context