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



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



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



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



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