Adding a row and a column

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

$$\textbf{K} = \begin{bmatrix}\textbf{A} & \textbf{b}\\ \textbf{b}^... ...^{-1} = \begin{bmatrix}\textbf{E} & \textbf{f}\\ \textbf{f}^T & g \end{bmatrix}$$

$$\Rightarrow \left\{\begin{array}{lcc}\textbf{A}\textbf{E} + \text... ...{b}g & = & \textbf{0}\\ \textbf{b}^T\textbf{f} + dg & = & 1 \end{array} \right.$$

$$\Rightarrow \textbf{K}^{-1} = \begin{bmatrix}\textbf{A}^{-1}(\tex... ...-\textbf{A}^{-1}\textbf{b}g\\ -(\textbf{A}^{-1}\textbf{b})^Tg & g \end{bmatrix}$$ (11)

with $g = (d - \textbf{b}^T\textbf{A}^{-1}\textbf{b})^{-1}$.