# Definition: Matrix Multiplication

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 $$B$$.

### Notes

• The matrix product of two matrices $$A$$ and $$B$$ is a matrix $$C$$, whose matrix elements in the $$i$$-th row and $$j$$-th column equal the dot products of the $$i$$-th row vector of $$A$$ with the $$j$$-th column vector of $$B$$, $$i=1,\ldots,m$$, $$j=1,\ldots,r$$.
• A matrix product is only possible if the number of rows of the first matrix equals the number of columns of the second matrix.
• In general, the matrix product is not commutative, i.e. $AB\neq BA.$

Definitions: 1
Examples: 2 3 4 5

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### References

#### Bibliography

1. Wille, D; Holz, M: "Repetitorium der Linearen Algebra", Binomi Verlag, 1994