证明下面的行列式式,谢谢了!设D(n)=|0 a(12) a(13). a(1n)|,证明当n为基数时,D(n)=0. |-a(12) 0 a(23).a(2n)| |-a(13) -a(23) 0.a(2n)| |. . . . .| |-a(1n) -a(2n) -a(3n)...0|
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![证明下面的行列式式,谢谢了!设D(n)=|0 a(12) a(13). a(1n)|,证明当n为基数时,D(n)=0. |-a(12) 0 a(23).a(2n)| |-a(13) -a(23) 0.a(2n)| |. . . . .| |-a(1n) -a(2n) -a(3n)...0|](/uploads/image/z/3945324-12-4.jpg?t=%E8%AF%81%E6%98%8E%E4%B8%8B%E9%9D%A2%E7%9A%84%E8%A1%8C%E5%88%97%E5%BC%8F%E5%BC%8F%2C%E8%B0%A2%E8%B0%A2%E4%BA%86%21%E8%AE%BED%28n%29%3D%7C0+a%2812%29+a%2813%29.+a%281n%29%7C%2C%E8%AF%81%E6%98%8E%E5%BD%93n%E4%B8%BA%E5%9F%BA%E6%95%B0%E6%97%B6%2CD%28n%29%3D0.+++++++++++++%7C-a%2812%29+0+a%2823%29.a%282n%29%7C+++++++++++++%7C-a%2813%29+-a%2823%29+0.a%282n%29%7C+++++++++++++%7C.+.+.+.+.%7C+++++++++++++%7C-a%281n%29+-a%282n%29+-a%283n%29...0%7C)
证明下面的行列式式,谢谢了!设D(n)=|0 a(12) a(13). a(1n)|,证明当n为基数时,D(n)=0. |-a(12) 0 a(23).a(2n)| |-a(13) -a(23) 0.a(2n)| |. . . . .| |-a(1n) -a(2n) -a(3n)...0|
证明下面的行列式式,谢谢了!
设D(n)=|0 a(12) a(13). a(1n)|,证明当n为基数时,D(n)=0.
|-a(12) 0 a(23).a(2n)|
|-a(13) -a(23) 0.a(2n)|
|. . . . .|
|-a(1n) -a(2n) -a(3n)...0|
证明下面的行列式式,谢谢了!设D(n)=|0 a(12) a(13). a(1n)|,证明当n为基数时,D(n)=0. |-a(12) 0 a(23).a(2n)| |-a(13) -a(23) 0.a(2n)| |. . . . .| |-a(1n) -a(2n) -a(3n)...0|
很容易拉,用行列式的性质——“行列式转置,其值不变”就行了.
证:根据行列式转置,行列式的值不变的性质
D(n)=|0 a(12) a(13).a(1n)| = |0 -a(12) -a(13).-a(1n)|
|-a(12) 0 a(23).a(2n)| |a(12) 0 -a(23).-a(2n)|
|-a(13) -a(23) 0.a(2n)| |a(13) a(23) 0 .-a(2n)|
|.....| |.....………|
|-a(1n) -a(2n) -a(3n)...0| |a(1n) a(2n) a(3n) ...… 0 |
=|0 a(12) a(13).a(1n)|
|-a(12) 0 a(23).a(2n)|
|-a(13) -a(23) 0.a(2n)| * (-1)^n (注释:这一步是把每行都提取一个(-1)出来)
|.....|
|-a(1n) -a(2n) -a(3n)...0|
换言之,Dn = Dn(-1)^n
当n为奇数时,Dn = - Dn.即,Dn = 0 (证毕)