1 determinant Q
2011-12-23 10:44 pm
(a). Prove that the determinant change sign if anytwo rows or columns are interchanged.(b). By (a), prove that if the element of a row (resp.a column) are identical to those ofanother row (resp. a column), then the valueof the determinant is zero.
回答 (3)
2012-01-03 5:18 am
✔ 最佳答案
(a) by induction(b) by (a)
2012-01-02 21:18:37 補充：
Let A=(C1, C2, ...,Ci, ...,Cj, ...,Cn), B=(C1, C2, ..., Cj,..., Ci, ..., Cn), where
Ci is a column vector(matrix of 1xn), i=1,2,3,...,n and Ci=Cj, then A=B.
If we interchange the i_th and j_th columns of A, then we get a new matrix B,
so
(1) det(A)= -det(B) (by (a)).
On the other hand, A=B (since Ci=Cj), so,
(2) det(A)=det(B).
From (1),(2), we know that det(A)=0.
Similarly, if there are two rows of A are identical, then det(A)=0.
2012-01-02 22:28:41 補充：
No concept of vectors are used in the above proof!
2012-01-02 11:15 pm
Hwo to (b) by (a) ?
2011-12-30 6:06 am
The prove of (a) maybe found in some pure maths/M2 textbooks.
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