![{\displaystyle \mathbf {A} ,\ \mathbf {B} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/fce31bf38e95c552da37dec581d6cad0f96cfe8f)
Summenform:
![{\displaystyle a_{rc},\ b_{rc}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a226dea830795b666e7fe6cb13e93b33d537d431)
Matrixform:
![{\displaystyle {\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}},\ {\begin{pmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0f2da495c3a910868e377d0ccd80202d8f57ee6)
![{\displaystyle \mathbf {0} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/62e8c650763635a93ddc69768c3c0c100afe985d)
Summenform:
![{\displaystyle 0_{rc}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b657026700540d37108863cf2d16fea517dc1a2c)
Matrixform:
![{\displaystyle {\begin{pmatrix}0&0&0\\0&0&0\\0&0&0\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f9bfee30b644f25ddef95454737f3b1ecd58057)
![{\displaystyle \mathbf {I} =\mathbf {1} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1cec98d3dedd52848fccd8cacaeb9d9d6d9a0d6)
Summenform:
![{\displaystyle i_{rc}={\begin{cases}1\ |\ r=c\\0\ |\ r\neq c\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d09046bf4a7eab64fca626ea94dc505a170fcadb)
![{\displaystyle i_{jj}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58da065a4eb356e4b42461ecbd8f5f996b8fa1aa)
Matrixform:
![{\displaystyle {\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cc04c39564230b6de6c1d727bd1227f7e6d7e72)
![{\displaystyle \mathbf {C} =s+\mathbf {A} =\mathbf {A} +s}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72ca1ab96950771ad30a4a5bdfdee096249f0060)
Summenform:
![{\displaystyle c_{rc}=s+a_{rc}=a_{rc}+s}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f457c5dc28e291306b212d22e7363a966da24038)
Matrixform:
![{\displaystyle {\begin{pmatrix}c_{11}&c_{12}\\c_{21}&c_{22}\\\end{pmatrix}}=s+{\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}={\begin{pmatrix}s+a_{11}&s+a_{12}\\s+a_{21}&s+a_{22}\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64abf2745ff39acd5293e8d5b72223e240673e06)
![{\displaystyle \mathbf {C} =\mathbf {A} +\mathbf {B} =\mathbf {B} +\mathbf {A} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/99b5abd65345d6a9c26c2ba5454d744baad001a5)
Summenform:
![{\displaystyle c_{rc}=a_{rc}+b_{rc}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be539fb6a45346978b3145897dd0ab4db22a22ad)
Matrixform:
![{\displaystyle {\begin{pmatrix}c_{11}&c_{12}\\c_{21}&c_{22}\end{pmatrix}}={\begin{pmatrix}a_{11}+b_{11}&a_{12}+b_{12}\\a_{21}+b_{21}&a_{22}+b_{22}\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0b837c1d494399c2431e669fbd5d8c75ee8a911)
![{\displaystyle \mathbf {C} =s\,\mathbf {A} =\mathbf {A} \,s}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dbf1373b096c3c70ea319d8df73ed78a9d4dca4)
Summenform:
![{\displaystyle c_{rc}=s\,a_{rc}=a_{rc}\,s}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e6d80fe4667be23ba10cd345867580ed4662854)
Matrixform:
![{\displaystyle {\begin{pmatrix}c_{11}&c_{12}\\c_{21}&c_{22}\end{pmatrix}}=s\,{\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}={\begin{pmatrix}s\,a_{11}&s\,a_{12}\\s\,a_{21}&s\,a_{22}\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70ba334536d0045aa3105eaa8bf3a80bd5c74c88)
![{\displaystyle \mathbf {C} =-\mathbf {A} =-1\,\mathbf {A} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7cfecf55131183f51bce8867b39ef2c23732c78)
Summenform:
![{\displaystyle c_{rc}=-1\,a_{rc}=-a_{rc}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c797c564f13e2f817e976605658dec4d08dac)
Matrixform:
![{\displaystyle {\begin{pmatrix}c_{11}&c_{12}\\c_{21}&c_{22}\end{pmatrix}}=-{\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}={\begin{pmatrix}-a_{11}&-a_{12}\\-a_{21}&-a_{22}\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2c14deb5160c4b36eaca0799c375c3c87e97f48)
![{\displaystyle \mathbf {C} =\mathbf {A} \,\mathbf {B} \neq \mathbf {B} \,\mathbf {A} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7905788483e11d68111c7f9c01226ef9650df0ed)
Summenform:
![{\displaystyle c_{rc}=\sum _{{\color {Red}k}=1}^{m}a_{r{\color {Red}k}}\,b_{{\color {Red}k}c}=a_{r{\color {Red}k}}\,b_{{\color {Red}k}c}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0de414f692a8c7218cdb5dd971328da724cf20de)
Matrixform:
![{\displaystyle {\begin{pmatrix}c_{11}&c_{12}&c_{13}\\c_{21}&c_{22}&c_{23}\\c_{31}&c_{32}&c_{33}\end{pmatrix}}={\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\a_{31}&a_{32}\end{pmatrix}}\,{\begin{pmatrix}b_{11}&b_{12}&b_{13}\\b_{21}&b_{22}&b_{23}\end{pmatrix}}=\dots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fc4b80f9967d19d2b6fb1289b2b05b24600a92e)
![{\displaystyle \dots =\sum _{{\color {Red}k}=1}^{3}{\begin{pmatrix}a_{1{\color {Red}k}}\,b_{{\color {Red}k}1}&a_{1{\color {Red}k}}\,b_{{\color {Red}k}2}&a_{1{\color {Red}k}}\,b_{{\color {Red}k}3}\\a_{2{\color {Red}k}}\,b_{{\color {Red}k}1}&a_{2{\color {Red}k}}\,b_{{\color {Red}k}2}&a_{2{\color {Red}k}}\,b_{{\color {Red}k}3}\\a_{3{\color {Red}k}}\,b_{{\color {Red}k}1}&a_{3{\color {Red}k}}\,b_{{\color {Red}k}2}&a_{3{\color {Red}k}}\,b_{{\color {Red}k}3}\\\end{pmatrix}}=\dots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d9c94572942d3fdd484605f630bc6a159007157)
![{\displaystyle \dots ={\begin{pmatrix}a_{1{\color {Red}1}}\,b_{{\color {Red}1}1}+a_{1{\color {Red}2}}\,b_{{\color {Red}2}1}+a_{1{\color {Red}3}}\,b_{{\color {Red}3}1}&a_{1{\color {Red}1}}\,b_{{\color {Red}1}2}+a_{1{\color {Red}2}}\,b_{{\color {Red}2}2}+a_{1{\color {Red}3}}\,b_{{\color {Red}3}2}&a_{1{\color {Red}1}}\,b_{{\color {Red}1}3}+a_{1{\color {Red}2}}\,b_{{\color {Red}2}3}+a_{1{\color {Red}3}}\,b_{{\color {Red}3}3}\\a_{2{\color {Red}1}}\,b_{{\color {Red}1}1}+a_{2{\color {Red}2}}\,b_{{\color {Red}2}1}+a_{2{\color {Red}3}}\,b_{{\color {Red}3}1}&a_{2{\color {Red}1}}\,b_{{\color {Red}1}2}+a_{2{\color {Red}2}}\,b_{{\color {Red}2}2}+a_{2{\color {Red}3}}\,b_{{\color {Red}3}2}&a_{2{\color {Red}1}}\,b_{{\color {Red}1}3}+a_{2{\color {Red}2}}\,b_{{\color {Red}2}3}+a_{2{\color {Red}3}}\,b_{{\color {Red}3}3}\\a_{3{\color {Red}1}}\,b_{{\color {Red}1}1}+a_{3{\color {Red}2}}\,b_{{\color {Red}2}1}+a_{3{\color {Red}3}}\,b_{{\color {Red}3}1}&a_{3{\color {Red}1}}\,b_{{\color {Red}1}2}+a_{3{\color {Red}2}}\,b_{{\color {Red}2}2}+a_{3{\color {Red}3}}\,b_{{\color {Red}3}2}&a_{3{\color {Red}1}}\,b_{{\color {Red}1}3}+a_{3{\color {Red}2}}\,b_{{\color {Red}2}3}+a_{3{\color {Red}3}}\,b_{{\color {Red}3}3}\\\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87e8224639df2302f03bd7bbb8c5fae2664ccc08)
![{\displaystyle \mathbf {C} =\mathbf {0} \,\mathbf {A} =\mathbf {A} \,\mathbf {0} =\mathbf {0} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef8ffc69af2984b02c1fbaa7a271da5f5c08e1cf)
Summenform:
![{\displaystyle c_{rc}=0\,a_{rc}=a_{rc}\,0=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b58d6853b8d9cafde2c07a7df28f2d7a9f885cd3)
![{\displaystyle \mathbf {C} =\mathbf {1} \,\mathbf {A} =\mathbf {A} \,\mathbf {1} =\mathbf {A} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/35f764cad3070b293c50ea846ec1ab996118889e)
Summenform:
![{\displaystyle c_{rc}=i_{rr}\,a_{rc}=a_{rc}\,i_{rr}=1\,a_{rc}=a_{rc}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89cf98e530ccb2aef1c92bd03b6aac3e2776471d)
![{\displaystyle \mathbf {A} \,\left(s\,\mathbf {A} \right)=\left(s\,\mathbf {A} \right)\,\mathbf {A} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/948cecc76b027da3c37ea2f1e666b0e65138aaf6)
Definition:
![{\displaystyle \mathbf {M} ^{-1}\,\mathbf {M} =\mathbf {M} \,\mathbf {M} ^{-1}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15d9ed1d1e38fc1e623708c8763c3652dbdb65e8)
Inverse ist eindeutig:
![{\displaystyle \left(\mathbf {L} \,\mathbf {M} =\mathbf {M} \,\mathbf {L} =1\right)\Rightarrow \mathbf {L} =\mathbf {M} ^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b26e551f5169d6c2886b67c5aaaacb5761296d56)
Rechenregeln:
![{\displaystyle \left(\mathbf {M} ^{-1}\right)^{-1}=\mathbf {M} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5da190fbfa6406636cf87dc92d347288be81d0e)
![{\displaystyle \left(\mathbf {M} \,\mathbf {N} \right)^{-1}=\left(\mathbf {M} ^{-1}\,\mathbf {N} ^{-1}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec3d064e2c3e96babd6d61d770a227ed52e084df)
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