(A|E )= $\left(\underset{A}{\underbrace{\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill -1\hfill \\ \hfill -8\hfill & \hfill 4\hfill & \hfill 1\hfill \\ \hfill -2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill \hfill \end{array}}}\right|\left.\underset{E}{\underbrace{\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill \hfill \end{array}}}\right)\begin{array}{c}\hfill \hfill \\ \hfill tauschen\hfill \\ \hfill tauschen\hfill \\ \hfill \hfill \end{array}$ .

$\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill -1\hfill \\ \hfill -2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill -8\hfill & \hfill 4\hfill & \hfill 1\hfill \\ \hfill \hfill \end{array}\right|\left.\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill \hfill \end{array}\right)\begin{array}{c}\hfill \hfill \\ \hfill \hfill \\ \hfill -4\cdot II\hfill \\ \hfill \hfill \end{array}$.
.
$\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill -1\hfill \\ \hfill -2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill \hfill \end{array}\right|\left.\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill -4\hfill \\ \hfill \hfill \end{array}\right)\begin{array}{c}\hfill +III\hfill \\ \hfill \hfill \\ \hfill \hfill \\ \hfill \hfill \end{array}$.
.

$\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill \hfill \end{array}\right|\left.\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill -4\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill -4\hfill \\ \hfill \hfill \end{array}\right)\begin{array}{c}\hfill \hfill \\ \hfill +2\cdot I\hfill \\ \hfill \hfill \\ \hfill \hfill \end{array}$.
.
Dies führt zu:

$\left(\underset{E}{\underbrace{\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}}}|\underset{A^{-1}}{\underbrace{\begin{array}{ccc}1& 1& -4\\ 2& 2& -7\\ 0& 1& -4\end{array}}}\right)$ $$$$$=(E|A^{-1})$

Die inverse Matrix $A^{-1}$ lautet damit: .
.
$A^{-1}=\left(\begin{array}{ccc}1& 1& -4\\ 2& 2& -7\\ 0& 1& -4\end{array}\right)$