Check der Beziehungen $A^{-1}A=\mathbf{1}$ und $AA^{-1}=\mathbf{1}$
Mit
$$A=\left(\begin{array}{cc}
A_{11} & A_{12}\\
A_{21} & A_{22}
\end{array}\right)$$
und
$$A^{-1}={1\over A_{11}A_{22}-A_{12}A_{21}}\left(\begin{array}{cc}
A_{22} & -A_{12}\\
-A_{21} & A_{11}
\end{array}\right)$$
ist
$$A^{-1}A={1\over A_{11}A_{22}-A_{12}A_{21}}
\left(\begin{array}{cc}
A_{11} & A_{12}\\
A_{21} & A_{22}
\end{array}\right)
\left(\begin{array}{cc}
A_{22} & -A_{12}\\
-A_{21} & A_{11}
\end{array}\right)=$$
$$={1\over A_{11}A_{22}-A_{12}A_{21}}\left(\begin{array}{cc}
A_{11}A_{22}-A_{12}A_{21} & -A_{11}A_{12}-A_{12}A_{11}\\
A_{21}A_{22}-A_{22}A_{21} & -A_{21}A_{12}+A_{22}A_{11}
\end{array}\right)=$$
$$={1\over A_{11}A_{22}-A_{12}A_{21}}\left(\begin{array}{cc}
A_{11}A_{22}-A_{12}A_{21} & 0\\
0 & A_{11}A_{22}-A_{12}A_{21}
\end{array}\right)=$$
$$=\left(\begin{array}{cc}
1 & 0\\
0 & 1
\end{array}\right)$$
und
$$AA^{-1}=
\left(\begin{array}{cc}
A_{22} & -A_{12}\\
-A_{21} & A_{11}
\end{array}\right){1\over A_{11}A_{22}-A_{12}A_{21}}
\left(\begin{array}{cc}
A_{11} & A_{12}\\
A_{21} & A_{22}
\end{array}\right)=$$
$$={1\over A_{11}A_{22}-A_{12}A_{21}}\left(\begin{array}{cc}
A_{22}A_{11}-A_{12}A_{21} & -A_{22}A_{12}-A_{12}A_{22}\\
-A_{21}A_{11}+A_{11}A_{21} & -A_{21}A_{12}+A_{11}A_{22}
\end{array}\right)=$$
$$={1\over A_{11}A_{22}-A_{12}A_{21}}\left(\begin{array}{cc}
A_{11}A_{22}-A_{12}A_{21} & 0\\
0 & A_{11}A_{22}-A_{12}A_{21}
\end{array}\right)=$$
$$=\left(\begin{array}{cc}
1 & 0\\
0 & 1
\end{array}\right)\,{\sf\small,}$$
womit die Beziehungen $A^{-1}A=\mathbf{1}$ und $AA^{-1}=\mathbf{1}$ überprüft sind.