Let \(A\in M_{m\times n}(F)\) and \(B\in M_{n\times r}(F)\) be two matrices. The matrix multiplication "$\cdot$" is defined by
\[ C:=AB=\pmatrix{ \alpha_{11} & \alpha_{12} & \ldots & \alpha_{1n} \cr \alpha_{21} & \alpha_{22} & \ldots & \alpha_{2n} \cr \vdots & \vdots & \ddots & \vdots \cr \alpha_{m1} & \alpha_{m2} & \ldots & \alpha_{mn} \cr }\cdot\pmatrix{ \beta_{11} & \beta_{12} & \ldots & \beta_{1r} \cr \beta_{21} & \beta_{22} & \ldots & \beta_{2r} \cr \vdots & \vdots & \ddots & \vdots \cr \beta_{n1} & \beta_{n2} & \ldots & \beta_{nr} \cr }=\pmatrix{ \sum_{k=1}^n\alpha_{1k}\beta_{k1} & \sum_{k=1}^n\alpha_{1k}\beta_{k2} & \ldots & \sum_{k=1}^n\alpha_{1k}\beta_{kr} \cr \sum_{k=1}^n\alpha_{2k}\beta_{k1} & \sum_{k=1}^n\alpha_{2k}\beta_{k2} & \ldots & \sum_{k=1}^n\alpha_{2k}\beta_{kr} \cr \vdots & \vdots & \ddots & \vdots \cr \sum_{k=1}^n\alpha_{mk}\beta_{k1} & \sum_{k=1}^n\alpha_{mk}\beta_{k2} & \ldots & \sum_{k=1}^n\alpha_{mk}\beta_{kr} \cr } \]
The resulting matrix $C\in M_{m\times r}(F)$ is called the matrix product of the two matrices \(A\) and \(B\).
Definitions: 1
Examples: 2 3 4 5
