ضرب ماتریسی - ویکی‌پدیا، دانشنامهٔ آزاد

${\displaystyle {\overset {3\times 4{\text{ matrix}}}{\begin{bmatrix}\cdot &\cdot &\cdot &\cdot \\\cdot &\cdot &\cdot &\cdot \\\color {Blue}1&\color {Blue}2&\color {Blue}3&\color {Blue}4\\\end{bmatrix}}}{\overset {4\times 5{\text{ matrix}}}{\begin{bmatrix}\cdot &\cdot &\cdot &\color {Red}a&\cdot \\\cdot &\cdot &\cdot &\color {Red}b&\cdot \\\cdot &\cdot &\cdot &\color {Red}c&\cdot \\\cdot &\cdot &\cdot &\color {Red}d&\cdot \\\end{bmatrix}}}={\overset {3\times 5{\text{ matrix}}}{\begin{bmatrix}\cdot &\cdot &\cdot &\cdot &\cdot \\\cdot &\cdot &\cdot &\cdot &\cdot \\\cdot &\cdot &\cdot &x_{3,4}&\cdot \\\end{bmatrix}}}}$

${\displaystyle x_{3,4}=({\color {Blue}1},{\color {Blue}2},{\color {Blue}3},{\color {Blue}4})\cdot ({\color {Red}a},{\color {Red}b},{\color {Red}c},{\color {Red}d})={\color {Blue}1}\times {\color {Red}a}+{\color {Blue}2}\times {\color {Red}b}+{\color {Blue}3}\times {\color {Red}c}+{\color {Blue}4}\times {\color {Red}d}}$.

${\displaystyle (AB)_{i,j}=\sum _{r=1}^{n}A_{i,r}B_{r,j}}$

${\displaystyle A\cdot B=A^{T}B={\begin{bmatrix}a_{1}&a_{2}&\cdots &a_{n}\end{bmatrix}}{\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{n}\end{bmatrix}}={\begin{bmatrix}a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}\end{bmatrix}}.}$.

${\displaystyle AB^{T}={\begin{bmatrix}a_{1}\\a_{2}\\\vdots \\a_{n}\end{bmatrix}}{\begin{bmatrix}b_{1}&b_{2}&\cdots &b_{n}\end{bmatrix}}={\begin{bmatrix}a_{1}b_{1}&a_{1}b_{2}&\cdots &a_{1}b_{n}\\a_{2}b_{1}&a_{2}b_{2}&\cdots &a_{2}b_{n}\\\vdots &\vdots &\ddots &\vdots \\a_{n}b_{1}&a_{n}b_{2}&\cdots &a_{n}b_{n}\\\end{bmatrix}}.}$

${\displaystyle \mathbf {A} ={\begin{bmatrix}{\color {Red}a_{1,1}}&{\color {Red}a_{1,2}}&\cdots &{\color {Red}a_{1,n}}\\{\color {ForestGreen}a_{2,1}}&{\color {ForestGreen}a_{2,2}}&\cdots &{\color {ForestGreen}a_{2,n}}\\\vdots &\vdots &\ddots &\vdots \\{\color {Blue}a_{m,1}}&{\color {Blue}a_{m,2}}&\cdots &{\color {Blue}a_{m,n}}\end{bmatrix}}={\begin{bmatrix}{\color {Red}A_{1}}\\{\color {ForestGreen}A_{2}}\\\vdots \\{\color {Blue}A_{m}}\end{bmatrix}}}$

${\displaystyle \mathbf {B} ={\begin{bmatrix}{\color {Red}b_{1,1}}&{\color {ForestGreen}b_{1,2}}&\cdots &{\color {Blue}b_{1,p}}\\{\color {Red}b_{2,1}}&{\color {ForestGreen}b_{2,2}}&\cdots &{\color {Blue}b_{2,p}}\\\vdots &\vdots &\ddots &\vdots \\{\color {Red}b_{n,1}}&{\color {ForestGreen}b_{n,2}}&\cdots &{\color {Blue}b_{n,p}}\end{bmatrix}}={\begin{bmatrix}{\color {Red}B_{1}}&{\color {ForestGreen}B_{2}}&\cdots &{\color {Blue}B_{p}}\end{bmatrix}}}$

${\displaystyle \mathbf {AB} ={\begin{bmatrix}A_{1}\\A_{2}\\\vdots \\A_{m}\end{bmatrix}}{\begin{bmatrix}B_{1}&B_{2}&\dots &B_{p}\end{bmatrix}}={\begin{bmatrix}(A_{1}\cdot B_{1})&(A_{1}\cdot B_{2})&\dots &(A_{1}\cdot B_{p})\\(A_{2}\cdot B_{1})&(A_{2}\cdot B_{2})&\dots &(A_{2}\cdot B_{p})\\\vdots &\vdots &\ddots &\vdots \\(A_{m}\cdot B_{1})&(A_{m}\cdot B_{2})&\dots &(A_{m}\cdot B_{p})\end{bmatrix}}.}$

${\displaystyle AB\neq BA}$

${\displaystyle \;\!\det(AB)=\det(BA)}$

${\displaystyle \ \mathbf {A} (\mathbf {BC} )=(\mathbf {AB} )\mathbf {C} }$

${\displaystyle \ \mathbf {A} (\mathbf {B} +\mathbf {C} )=\mathbf {AB} +\mathbf {AC} }$

${\displaystyle \ (\mathbf {A} +\mathbf {B} )\mathbf {C} =\mathbf {AC} +\mathbf {BC} }$.

${\displaystyle \ c(\mathbf {AB} )=(c\mathbf {A} )\mathbf {B} }$

${\displaystyle \ (\mathbf {A} c)\mathbf {B} =\mathbf {A} (c\mathbf {B} )}$

${\displaystyle \ (\mathbf {AB} )c=\mathbf {A} (\mathbf {B} c)}$

${\displaystyle (r\mathbf {A} )_{ij}=r\cdot a_{ij}.\,}$

${\displaystyle \mathbf {A} ={\begin{bmatrix}a&b\\c&d\end{bmatrix}}}$

${\displaystyle r\cdot \mathbf {A} ={\begin{bmatrix}r\cdot a&r\cdot b\\r\cdot c&r\cdot d\end{bmatrix}}}$