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
$$ (A+B)+C=A+(B+C) $$
$$ \left( \left( \begin{array}{cccc}a_{11}&a_{12}&\cdots&a_{1m}\\a_{21}&a_{22}&\cdots&a_{2m}\\ \vdots&\vdots&\ddots&\vdots\\a_{n1}&a_{n2}&\cdots&a_{nm}\end{array} \right) + \left( \begin{array}{cccc}b_{11}&b_{12}&\cdots&b_{1m}\\b_{21}&b_{22}&\cdots&b_{2m}\\ \vdots&\vdots&\ddots&\vdots\\b_{n1}&b_{n2}&\cdots&b_{nm}\end{array} \right) \right) + \left( \begin{array}{cccc}c_{11}&c_{12}&\cdots&c_{1m}\\c_{21}&c_{22}&\cdots&c_{2m}\\ \vdots&\vdots&\ddots&\vdots\\c_{n1}&c_{n2}&\cdots&c_{nm}\end{array} \right) $$ $$ = \left( \begin{array}{cccc}a_{11}&a_{12}&\cdots&a_{1m}\\a_{21}&a_{22}&\cdots&a_{2m}\\ \vdots&\vdots&\ddots&\vdots\\a_{n1}&a_{n2}&\cdots&a_{nm}\end{array} \right) + \left( \left( \begin{array}{cccc}b_{11}&b_{12}&\cdots&b_{1m}\\b_{21}&b_{22}&\cdots&b_{2m}\\ \vdots&\vdots&\ddots&\vdots\\b_{n1}&b_{n2}&\cdots&b_{nm}\end{array} \right) + \left( \begin{array}{cccc}c_{11}&c_{12}&\cdots&c_{1m}\\c_{21}&c_{22}&\cdots&c_{2m}\\ \vdots&\vdots&\ddots&\vdots\\c_{n1}&c_{n2}&\cdots&c_{nm}\end{array} \right) \right) $$
$$ => \left( \begin{array}{cccc}a_{11}+b_{11}&a_{12}+b_{12}&\cdots&a_{1m}+b_{1m}\\a_{21}+b_{21}&a_{22}+b_{22}&\cdots&a_{2m}+b_{2m}\\ \vdots&\vdots&\ddots&\vdots\\a_{n1}+b_{n1}&a_{n2}+b_{n2}&\cdots&a_{nm}+b_{nm}\end{array} \right) + \left( \begin{array}{cccc}c_{11}&c_{12}&\cdots&c_{1m}\\c_{21}&c_{22}&\cdots&c_{2m}\\ \vdots&\vdots&\ddots&\vdots\\c_{n1}&c_{n2}&\cdots&c_{nm}\end{array} \right) $$ $$ = \left( \begin{array}{cccc}a_{11}&a_{12}&\cdots&a_{1m}\\a_{21}&a_{22}&\cdots&a_{2m}\\ \vdots&\vdots&\ddots&\vdots\\a_{n1}&a_{n2}&\cdots&a_{nm}\end{array} \right) + \left( \begin{array}{cccc}b_{11}+c_{11}&b_{12}+c_{12}&\cdots&b_{1m}+c_{1m}\\b_{21}+c_{21}&b_{22}+c_{22}&\cdots&b_{2m}+c_{2m}\\ \vdots&\vdots&\ddots&\vdots\\b_{n1}+c_{n1}&b_{n2}+c_{n2}&\cdots&b_{nm}+c_{nm}\end{array} \right) $$
$$ => \left( \begin{array}{cccc}a_{11}+b_{11}+c_{11}&a_{12}+b_{12}+c_{12}&\cdots&a_{1m}+b_{1m}+c_{1m}\\a_{21}+b_{21}+c_{21}&a_{22}+b_{22}+c_{22}&\cdots&a_{2m}+b_{2m}+c_{2m}\\ \vdots&\vdots&\ddots&\vdots\\a_{n1}+b_{n1}+c_{n1}&a_{n2}+b_{n2}+c_{n2}&\cdots&a_{nm}+b_{nm}+c_{nm}\end{array} \right) $$ $$ = \left( \begin{array}{cccc}a_{11}+b_{11}+c_{11}&a_{12}+b_{12}+c_{12}&\cdots&a_{1m}+b_{1m}+c_{1m}\\a_{21}+b_{21}+c_{21}&a_{22}+b_{22}+c_{22}&\cdots&a_{2m}+b_{2m}+c_{2m}\\ \vdots&\vdots&\ddots&\vdots\\a_{n1}+b_{n1}+c_{n1}&a_{n2}+b_{n2}+c_{n2}&\cdots&a_{nm}+b_{nm}+c_{nm}\end{array} \right) $$
Da bei auf beiden Seiten das gleiche Ergebnis herauskommt, stimmt unsere These, dass Matrizen assoziativ sind.
qed