Die Determinante der Matrix ist

${\displaystyle {}{\frac {x+1}{x^{2}}}-x^{3}={\frac {-x^{5}+x+1}{x^{2}}}\,,}$

die inverse Matrix ist daher

${\displaystyle {}{\begin{aligned}{\frac {x^{2}}{-x^{5}+x+1}}{\begin{pmatrix}{\frac {x+1}{x^{2}}}&-x\\-x^{2}&1\end{pmatrix}}&={\begin{pmatrix}{\frac {x^{2}}{-x^{5}+x+1}}\cdot {\frac {x+1}{x^{2}}}&-{\frac {x^{2}}{-x^{5}+x+1}}\cdot x\\-{\frac {x^{2}}{-x^{5}+x+1}}\cdot x^{2}&{\frac {x^{2}}{-x^{5}+x+1}}\end{pmatrix}}\\&={\begin{pmatrix}{\frac {x+1}{-x^{5}+x+1}}&-{\frac {x^{3}}{-x^{5}+x+1}}\\-{\frac {x^{4}}{-x^{5}+x+1}}&{\frac {x^{2}}{-x^{5}+x+1}}\end{pmatrix}}.\end{aligned}}}$