II Addition von Matrizen ist assoziativ
These: Die Addition von Matrizen ist assoziativ, d.h. (A+B)+C=A+(B+C)
Beweis: Allgemein mit (n,m)-Matrizen. Wir rechnen einfach:
$$ (A+B)+C= \left( \left( \begin{array}{cccc}a_{11}&a_{12}&\cdots&a_{1m}\\a_{21}&a_{22}&\cdots&a_{2m}\\ \vdots&\vdots&\ddots&\vdots\\a_{n1}&a_{n2}&\cdots&a_{nm}\end{array} \right) + \left( \begin{array}{cccc}b_{11}&b_{12}&\cdots&b_{1m}\\b_{21}&b_{22}&\cdots&b_{2m}\\ \vdots&\vdots&\ddots&\vdots\\b_{n1}&b_{n2}&\cdots&b_{nm}\end{array} \right) \right) + \left( \begin{array}{cccc}c_{11}&c_{12}&\cdots&c_{1m}\\c_{21}&c_{22}&\cdots&c_{2m}\\ \vdots&\vdots&\ddots&\vdots\\c_{n1}&c_{n2}&\cdots&c_{nm}\end{array} \right) $$
$$ =\left( \begin{array}{cccc}a_{11}+b_{11}&a_{12}+b_{12}&\cdots&a_{1m}+b_{1m}\\a_{21}+b_{21}&a_{22}+b_{22}&\cdots&a_{2m}+b_{2m}\\ \vdots&\vdots&\ddots&\vdots\\a_{n1}+b_{n1}&a_{n2}+b_{n2}&\cdots&a_{nm}+b_{nm}\end{array} \right) + \left( \begin{array}{cccc}c_{11}&c_{12}&\cdots&c_{1m}\\c_{21}&c_{22}&\cdots&c_{2m}\\ \vdots&\vdots&\ddots&\vdots\\c_{n1}&c_{n2}&\cdots&c_{nm}\end{array} \right) $$ $$ = \left( \begin{array}{cccc}(a_{11}+b_{11})+c_{11}&(a_{12}+b_{12})+c_{12}&\cdots&(a_{1m}+b_{1m})+c_{1m}\\(a_{21}+b_{21})+c_{21}&(a_{22}+b_{22})+c_{22}&\cdots&(a_{2m}+b_{2m})+c_{2m}\\ \vdots&\vdots&\ddots&\vdots\\(a_{n1}+b_{n1})+c_{n1}&(a_{n2}+b_{n2})+c_{n2}&\cdots&(a_{nm}+b_{nm})+c_{nm}\end{array} \right) $$
$$ = \left( \begin{array}{cccc}a_{11}+(b_{11}+c_{11})&a_{12}+(b_{12}+c_{12})&\cdots&a_{1m}+(b_{1m}+c_{1m})\\a_{21}+(b_{21}+c_{21})&a_{22}+(b_{22}+c_{22})&\cdots&a_{2m}+(b_{2m}+c_{2m})\\ \vdots&\vdots&\ddots&\vdots\\a_{n1}+(b_{n1}+c_{n1})&a_{n2}+(b_{n2}+c_{n2})&\cdots&a_{nm}+(b_{nm}+c_{nm})\end{array} \right) $$
$$ = \left( \begin{array}{cccc}a_{11}&a_{12}&\cdots&a_{1m}\\a_{21}&a_{22}&\cdots&a_{2m}\\ \vdots&\vdots&\ddots&\vdots\\a_{n1}&a_{n2}&\cdots&a_{nm}\end{array} \right) + \left( \begin{array}{cccc}b_{11}+c_{11}&b_{12}+c_{12}&\cdots&b_{1m}+c_{1m}\\b_{21}+c_{21}&b_{22}+c_{22}&\cdots&b_{2m}+c_{2m}\\ \vdots&\vdots&\ddots&\vdots\\b_{n1}+c_{n1}&b_{n2}+c_{n2}&\cdots&b_{nm}+c_{nm}\end{array} \right) $$
$$ = \left( \begin{array}{cccc}a_{11}&a_{12}&\cdots&a_{1m}\\a_{21}&a_{22}&\cdots&a_{2m}\\ \vdots&\vdots&\ddots&\vdots\\a_{n1}&a_{n2}&\cdots&a_{nm}\end{array} \right) + \left( \left( \begin{array}{cccc}b_{11}&b_{12}&\cdots&b_{1m}\\b_{21}&b_{22}&\cdots&b_{2m}\\ \vdots&\vdots&\ddots&\vdots\\b_{n1}&b_{n2}&\cdots&b_{nm}\end{array} \right) + \left( \begin{array}{cccc}c_{11}&c_{12}&\cdots&c_{1m}\\c_{21}&c_{22}&\cdots&c_{2m}\\ \vdots&\vdots&\ddots&\vdots\\c_{n1}&c_{n2}&\cdots&c_{nm}\end{array} \right) \right) $$ $$=A+(B+C) $$
qed