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# 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.)

感謝，請來領點數

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

- heavenLv 69 years agoFavorite Answer
其實你把B轉一轉就會變成A了

就像

12

34

與

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

- 老怪物Lv 79 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.