$\left(A|c\right)=\left(\begin{array}{ccc}\hfill 4\hfill & \hfill -1\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill -1\hfill & \hfill 2\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill \hfill \end{array}\right|\left.\begin{array}{c}\hfill 6\hfill \\ \hfill 0\hfill \\ \hfill 2\hfill \\ \hfill 3\hfill \\ \hfill \hfill \end{array}\right)\begin{array}{c}\hfill tauschen\hfill \\ \hfill tauschen\hfill \\ \hfill \hfill \\ \hfill \hfill \\ \hfill \hfill \end{array}\Rightarrow$.

$\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill 4\hfill & \hfill -1\hfill & \hfill -1\hfill \\ \hfill -1\hfill & \hfill 2\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill \hfill \end{array}\right|\left.\begin{array}{c}\hfill 0\hfill \\ \hfill 6\hfill \\ \hfill 2\hfill \\ \hfill 3\hfill \\ \hfill \hfill \end{array}\right)\begin{array}{c}\hfill \hfill \\ \hfill -4\cdot I\hfill \\ \hfill +I\hfill \\ \hfill -3\cdot I\hfill \\ \hfill \hfill \end{array}\Rightarrow$.

$\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill -1\hfill & \hfill -9\hfill \\ \hfill 0\hfill & \hfill 2\hfill & \hfill 4\hfill \\ \hfill 0\hfill & \hfill -1\hfill & \hfill -6\hfill \end{array}\right|\left.\begin{array}{c}\hfill 0\hfill \\ \hfill 6\hfill \\ \hfill 2\hfill \\ \hfill 3\hfill \\ \hfill \hfill \end{array}\right)\begin{array}{c}\hfill \hfill \\ \hfill \hfill \\ \hfill +2\cdot II\hfill \\ \hfill -II\hfill \\ \hfill \hfill \end{array}\Rightarrow$.

$\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill -1\hfill & \hfill -9\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -14\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 3\hfill \\ \hfill \hfill \end{array}\right|\left.\begin{array}{c}\hfill 0\hfill \\ \hfill 6\hfill \\ \hfill 14\hfill \\ \hfill -3\hfill \\ \hfill \hfill \end{array}\right)\begin{array}{c}\hfill \hfill \\ \hfill \hfill \\ \hfill ∕14\hfill \\ \hfill ∕3\hfill \\ \hfill \hfill \end{array}\Rightarrow$.

$\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill -1\hfill & \hfill -9\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill \hfill \end{array}\right|\left.\begin{array}{c}\hfill 0\hfill \\ \hfill 6\hfill \\ \hfill 1\hfill \\ \hfill -1\hfill \\ \hfill \hfill \end{array}\right)\begin{array}{c}\hfill \hfill \\ \hfill \hfill \\ \hfill \hfill \\ \hfill +III\hfill \\ \hfill \hfill \end{array}\Rightarrow$.

$\underset{A^{*}}{\underbrace{\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill -1\hfill & \hfill -9\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \hfill \end{array}\right|}}\underset{c^{*}}{\underbrace{\left.\begin{array}{c}\hfill 0\hfill \\ \hfill 6\hfill \\ \hfill 1\hfill \\ \hfill 0\hfill \\ \hfill \hfill \end{array}\right)}}=\left(A^{*}|c^{*}\right)$.
Daraus folgt: $rg\left(A\right)=rg\left(A^{*}\right)=3,rg\left(A|c\right)=rg\left(A^{*}|c^{*}\right)=3$. .
Aus dem gestaffelten System kann man das Ergebnis ablesen: .
$x_3=-1$ .
$-x_{-}2-9x_3=6\Rightarrow x_2=3$ .
$x_1+2x_3=0\Rightarrow x_1=2$. .