${{{A}} {{x}}} = {b}$
${ {\left[ \begin{matrix} 3 & 2 & -1 \\ 2 & -2 & 4 \\ -1 & \frac{1}{2} & -1\end{matrix} \right]} {\left[ \begin{matrix} {x_{1,1}} \\ {x_{2,1}} \\ {x_{3,1}}\end{matrix} \right]}} = {\left[ \begin{matrix} 1 \\ -2 \\ 0\end{matrix} \right]}$
${{A}^{-1}} = {\left[ \begin{matrix} 0 & -{\frac{1}{2}} & -{2} \\ \frac{2}{3} & \frac{4}{3} & \frac{14}{3} \\ \frac{1}{3} & \frac{7}{6} & \frac{10}{3}\end{matrix} \right]}$ extra $\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix} \right]$ message nil
$A^{-1} \cdot b =$ $\left[ \begin{matrix} 1 \\ -{2} \\ -{2}\end{matrix} \right]$
${A}^{-1}$ $(b)=$ $\left[ \begin{matrix} 1 \\ -{2} \\ -{2}\end{matrix} \right]$ extra $\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix} \right]$ message nil
${{{{A}^{-1}}} {{A}}} = {\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix} \right]}$



${{{A}} {{x}}} = {b}$
${ {\left[ \begin{matrix} 2 & 3 \\ 4 & 9\end{matrix} \right]} {\left[ \begin{matrix} {x_{1,1}} \\ {x_{2,1}}\end{matrix} \right]}} = {\left[ \begin{matrix} 6 \\ 15\end{matrix} \right]}$
${{A}^{-1}} = {\left[ \begin{matrix} \frac{3}{2} & -{\frac{1}{2}} \\ -{\frac{2}{3}} & \frac{1}{3}\end{matrix} \right]}$ extra $\left[ \begin{matrix} 1 & 0 \\ 0 & 1\end{matrix} \right]$ message nil
$A^{-1} \cdot b =$ $\left[ \begin{matrix} \frac{3}{2} \\ 1\end{matrix} \right]$
${A}^{-1}$ $(b)=$ $\left[ \begin{matrix} \frac{3}{2} \\ 1\end{matrix} \right]$ extra $\left[ \begin{matrix} 1 & 0 \\ 0 & 1\end{matrix} \right]$ message nil
${{{{A}^{-1}}} {{A}}} = {\left[ \begin{matrix} 1 & 0 \\ 0 & 1\end{matrix} \right]}$



${{{A}} {{x}}} = {b}$
${ {\left[ \begin{matrix} 1 & -2 \\ 3 & 5 \\ 4 & 3\end{matrix} \right]} {\left[ \begin{matrix} {x_{1,1}} \\ {x_{2,1}}\end{matrix} \right]}} = {\left[ \begin{matrix} -1 \\ 8 \\ 7\end{matrix} \right]}$
${{A}^{-1}} = {\left[ \begin{matrix} \frac{5}{11} & \frac{2}{11} & 0 \\ -{\frac{3}{11}} & \frac{1}{11} & 0 \\ -{1} & -{1} & 1\end{matrix} \right]}$ extra $\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 0\end{matrix} \right]$ message system is overconstrained
$A^{-1} \cdot b =$ $\left[ \begin{matrix} 1 \\ 1 \\ 0\end{matrix} \right]$
${A}^{-1}$ $(b)=$ $\left[ \begin{matrix} 1 \\ 1 \\ 0\end{matrix} \right]$ extra $\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 0\end{matrix} \right]$ message nil
${{{{A}^{-1}}} {{A}}} = {\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 0\end{matrix} \right]}$



${{{A}} {{x}}} = {b}$
${ {\left[ \begin{matrix} 3 & 2 \\ 3 & 2\end{matrix} \right]} {\left[ \begin{matrix} {x_{1,1}} \\ {x_{2,1}}\end{matrix} \right]}} = {\left[ \begin{matrix} 6 \\ 12\end{matrix} \right]}$
${{A}^{-1}} = {\left[ \begin{matrix} \frac{1}{3} & 0 \\ -{1} & 1\end{matrix} \right]}$ extra $\left[ \begin{matrix} 1 & \frac{2}{3} \\ 0 & 0\end{matrix} \right]$ message nil
$A^{-1} \cdot b =$ $\left[ \begin{matrix} 2 \\ 6\end{matrix} \right]$
${A}^{-1}$ $(b)=$ $\left[ \begin{matrix} 2 \\ 6\end{matrix} \right]$ extra $\left[ \begin{matrix} 1 & \frac{2}{3} \\ 0 & 0\end{matrix} \right]$ message nil
${{{{A}^{-1}}} {{A}}} = {\left[ \begin{matrix} 1 & \frac{2}{3} \\ 0 & 0\end{matrix} \right]}$



${{{A}} {{x}}} = {b}$
${ {\left[ \begin{matrix} 1 & 1 \\ 2 & 1 \\ 3 & 2\end{matrix} \right]} {\left[ \begin{matrix} {x_{1,1}} \\ {x_{2,1}}\end{matrix} \right]}} = {\left[ \begin{matrix} 1 \\ 1 \\ 3\end{matrix} \right]}$
${{A}^{-1}} = {\left[ \begin{matrix} -{1} & 1 & 0 \\ 2 & -{1} & 0 \\ -{1} & -{1} & 1\end{matrix} \right]}$ extra $\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 0\end{matrix} \right]$ message system is overconstrained
$A^{-1} \cdot b =$ $\left[ \begin{matrix} 0 \\ 1 \\ 1\end{matrix} \right]$
${A}^{-1}$ $(b)=$ $\left[ \begin{matrix} 0 \\ 1 \\ 1\end{matrix} \right]$ extra $\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 0\end{matrix} \right]$ message system is overconstrained
${{{{A}^{-1}}} {{A}}} = {\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 0\end{matrix} \right]}$