行列式的性质 上三角矩阵的行列式等于其对角元之积: 0a22 a11a22…··amn 证明:对一个全排列i…in,ai1a2…ahnm中有可能非零的项只能 是in=n,in-1=n-1,…,i=1,而此时ii2…in就是自然顺序, a11a12 0 所以N(i12…in)=0。排列定义 ∑(-1)N(2nlan112…an中可能非零的项仅有 为12…的全排列 项a1122…·am,所以有欲证结论。pê E￾ŒÆ£Á‘¤ 1ª 1ª5Ÿ 1ª5Ÿ þnÝ 1ªuÙéƒÈµ          a11 a12 ⋅ ⋅ ⋅ a1n 0 a22 ⋅ ⋅ ⋅ a2n . . . . . . . . . . . . 0 0 ⋅ ⋅ ⋅ ann          = a11a22 ⋅ ⋅ ⋅ ann" y²µ é‡ü i1 i2 ⋅ ⋅ ⋅ in§ai11ai22 ⋅ ⋅ ⋅ ainn ¥kŒUš"‘U ´ in = n, in−1 = n − 1, . . . , i1 = 1§ dž i1i2 ⋅ ⋅ ⋅ in Ò´g,^S§ ¤± N(i1i2 ⋅ ⋅ ⋅ in) = 0"ü½Â          a11 a12 ⋅ ⋅ ⋅ a1n 0 a22 ⋅ ⋅ ⋅ a2n . . . . . . . . . . . . 0 0 ⋅ ⋅ ⋅ ann          = ∑ i1 i2⋅⋅⋅in 12⋅⋅⋅nü (−1) N(i1 i2⋅⋅⋅in) ai11ai22 ⋅ ⋅ ⋅ ainn ¥ŒUš"‘=k ‘ a11a22 ⋅ ⋅ ⋅ ann§¤±ky(Ø" □