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)