# examples of non-commutative operations

A standard example of a non-commutative operation is matrix multiplication^{}. Consider the following two integer matrices:

$$A=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right),B=\left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)$$ |

If we compute $AB$ we get

$$AB=\left(\begin{array}{cc}\hfill 0\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)$$ |

but if we compute $BA$ we have

$$BA=\left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right).$$ |

Since $AB\ne BA$ we conclude that matrix product is not commutative^{}.

Operations do not necessarily have to operate on numbers. Another classic example is function composition. Let $f$ and $g$ be real functions given by

$$f(x)={x}^{2},g(x)=2x.$$ |

We see that

$$(f\circ g)(x)=f(g(x))={(2x)}^{2}=4{x}^{2},$$ |

but

$$(g\circ f)(x)=g(f(x))=2({x}^{2})=2{x}^{2}.$$ |

Since we got different functions^{}, we conclude that function composition is not commutative.

Title | examples of non-commutative operations |
---|---|

Canonical name | ExamplesOfNoncommutativeOperations |

Date of creation | 2013-03-22 15:03:04 |

Last modified on | 2013-03-22 15:03:04 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 10 |

Author | yark (2760) |

Entry type | Example |

Classification | msc 20-00 |