Removing the first row and column

From a given non-singular matrix $\textbf{K}$ a row and column are removed as shown below, resulting in matrix $\textbf{D}$. The inverse matrix $\textbf{D}^{-1}$ can then easily be expressed in terms of the known elements of $\textbf{K}^{-1}$ as follows:

$$\textbf{K} = \begin{bmatrix}a & \textbf{b}^T\\ \textbf{b} & \text... ...}^{-1} = \begin{bmatrix}e & \textbf{f}^T\\ \textbf{f} & \textbf{G}\end{bmatrix}$$

$$\Rightarrow \left\{\begin{array}{lcc} \textbf{b}e + \textbf{D}\te... ...xtbf{b}\textbf{f}^T + \textbf{D}\textbf{G} & = & \textbf{I} \end{array} \right.$$

$$\Rightarrow \textbf{D}^{-1} = \textbf{G} - \textbf{f}\textbf{f}^T/e.$$ (12)