Lösung für Aufgabe 5.3.4
Rechnen Sie nach, dass in $(M_{2}(\R),+,\cdot)$ die Distributivgesetze gelten.Es gilt
\begin{eqnarray*}
A(B+C) &=&
\begin{pmatrix}
a_{11}& a_{12}\\a_{21}& a_{22}
\end{pmatrix}
\begin{pmatrix}
b_{11} + c_{11}& b_{12} + c_{12}\\b_{21} + c_{21}& b_{22} + c_{22}
\end{pmatrix}
=
\begin{pmatrix}
a_{11} (b_{11} + c_{11}) + a_{12} (b_{21} + c_{21})& a_{11} (b_{12} + c_{12}) + a_{12} (b_{22} + c_{22})\\a_{21} (b_{11} + c_{11}) + a_{22} (b_{21} + c_{21})& a_{21} (b_{12} + c_{12}) + a_{22} (b_{22} + c_{22})
\end{pmatrix}\\
&=&
\begin{pmatrix}
a_{11} b_{11} + a_{12} b_{21} + a_{11} c_{11} + a_{12} c_{21}& a_{11} b_{12} + a_{12} b_{22} + a_{11} c_{12} + a_{12} c_{22}\\a_{21} b_{11} + a_{22} b_{21} + a_{21} c_{11} + a_{22} c_{21}& a_{21} b_{12} + a_{22} b_{22} + a_{21} c_{12} + a_{22} c_{22}
\end{pmatrix}\\
&=&
\begin{pmatrix}
a_{11} b_{11} + a_{12} b_{21}& a_{11} b_{12} + a_{12} b_{22}\\a_{21} b_{11} + a_{22} b_{21}& a_{21} b_{12} + a_{22} b_{22}
\end{pmatrix}+
\begin{pmatrix}
a_{11} c_{11} + a_{12} c_{21}& a_{11} c_{12} + a_{12} c_{22}\\a_{21} c_{11} + a_{22} c_{21}& a_{21} c_{12} + a_{22} c_{22}
\end{pmatrix}=
AB+AC\\
(A+B)C &=&
\begin{pmatrix}
a_{11} + b_{11}& a_{12} + b_{12}\\a_{21} + b_{21}& a_{22} + b_{22}
\end{pmatrix}
\begin{pmatrix}
c_{11}& c_{12}\\c_{21}& c_{22}
\end{pmatrix}=
\begin{pmatrix}
(a_{11} + b_{11}) c_{11} + (a_{12} + b_{12}) c_{21}& (a_{11} + b_{11}) c_{12} + (a_{12} + b_{12}) c_{22}\\(a_{21} + b_{21}) c_{11} + (a_{22} + b_{22}) c_{21}& (a_{21} + b_{21}) c_{12} + (a_{22} + b_{22}) c_{22}
\end{pmatrix}\\
&=&
\begin{pmatrix}
a_{11} c_{11} + b_{11} c_{11} + a_{12} c_{21} + b_{12} c_{21}& a_{11} c_{12} + b_{11} c_{12} + a_{12} c_{22} + b_{12} c_{22}\\a_{21} c_{11} + b_{21} c_{11} + a_{22} c_{21} + b_{22} c_{21}& a_{21} c_{12} + b_{21} c_{12} + a_{22} c_{22} + b_{22} c_{22}
\end{pmatrix}\\
&=&
\begin{pmatrix}
a_{11} c_{11} + a_{12} c_{21}& a_{11} c_{12} + a_{12} c_{22}\\a_{21} c_{11} + a_{22} c_{21}& a_{21} c_{12} + a_{22} c_{22}
\end{pmatrix}+
\begin{pmatrix}
b_{11} c_{11} + b_{12} c_{21}& b_{11} c_{12} + b_{12} c_{22}\\b_{21} c_{11} + b_{22} c_{21}& b_{21} c_{12} + b_{22} c_{22}
\end{pmatrix} = AC+BC
\end{eqnarray*}