匿名
匿名 發問於 科學及數學數學 · 8 年前

Matrix Question

Prove that the following determinant equals zero without expanding the determinant

︱a-b b-c c-a︱

︱b-c c-a a-b︱

︱c-a a-b b-c︱

2 個解答

評分
  • 8 年前
    最愛解答

    Prove that the following determinant equalszero without expanding the determinant

    |a-b b-c c-a|

    |b-c c-a a-b|

    |c-a a-b b-c|

    Sol|a-b b-c c-a|*1──────

    |b-c c-a a-b|<───────|

    |c-a a-b b-c|

     |    a-b     b-c     c-a|

    =|a-b+b-c b-c+c-a c-a+a-b|

     |    c-a     a-b     b-c|

     |a-b b-c c-a|

    =|a-c b-a c-b|*1──────

     |c-a a-b b-c|<───────|

     |    a-b     b-c     c-a|

    =|    a-c     b-a     c-b|

     |a-c+c-a b-a+a-b c-b+b-c|

     |a-b b-c c-a|

    =|a-c b-a c-b|

     |  0   0   0|

    =0

  • wy
    Lv 7
    8 年前

    | a - b b - c c - a |

    | b - c c - a a - b |

    | c - a a - b b - c |

    = | a - b a - c c - a | C1 + C2 to C2

    | b - c b - a a - b |

    | c - a c - b b - c |

    = | a - b a - c 0 | C2 + C3 to C3

    | b - c b - a 0 |

    | c - a c - b 0 |

    = 0.

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