天天happy asked in 科學數學 · 9 years ago

線性代數矩陣證明問題

Prove that if A is an invertible matrix and B is row equivalent to A,then B is also invertible.

(Matrices A and B are said to be row equivalent if either can be obtained from the other by a sequence of elementary row operations.)

Update:

感謝,請來領點數

Update 2:

還有一題

Let Ax=0 be a homogeneous system of n linear equations in n unknowns that has only the trivial solution.Show that if k is any positive integer,then the system (A^k)x=0 also has only the trivial solution.

3 Answers

Rating
  • heaven
    Lv 6
    9 years ago
    Favorite Answer

    其實你把B轉一轉就會變成A了

    就像

    12

    34

    13

    24

    你頭轉一轉就會知他們是一樣的東西

    2011-10-17 23:58:25 補充:

    剛剛那例子怪怪的

    不過以3x3為例

    要求determine....你會計算出6個項目

    B與A的項目都是相同

    所以兩者determine一樣

  • 9 years ago

    B is row equivalent to A iff.

    for a sequence of elementary matrices E1,...,En,

    B = (En)...(E1)A.

    So B^{-1} = A^{-1}(E1)^{-1}...(En)^{-1}.

    2011-10-14 20:30:52 補充:

    Ax = 0 has only the trivial solution, means that A is full rank,

    or equivalently, A is nonsingular.

    Therefore, for any positive integer k, A^k is nonsingular too.

    Then, A^k X = 0 has only the trivial solution.

  • 9 years ago

    Elementary matrix

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