行列式的性质 下三角矩阵的行列式等于其对角元之积: 0 n21a22 0 an 证明:对一个全排列ii…in,a142…ainm中有可能非零的项只能 是ⅱ1=1,=2,…,i=n,而此时i1i…i就是自然顺序,所 a2 0 以N(i2…in)=0。排列定义 净(-1)A(2)an2“中可能非零的项仅有 的全排列 a12…am,所以有欲证结论。pê E￾ŒÆ£Á‘¤ 1ª 1ª5Ÿ 1ª5Ÿ enÝ 1ªuÙéƒÈµ          a11 0 ⋅ ⋅ ⋅ 0 a21 a22 ⋅ ⋅ ⋅ 0 . . . . . . . . . . . . an1 an2 ⋅ ⋅ ⋅ ann          = a11a22 ⋅ ⋅ ⋅ ann" y²µ é‡ü i1i2 ⋅ ⋅ ⋅ in§ai11ai22 ⋅ ⋅ ⋅ ainn ¥kŒUš"‘U ´ i1 = 1, i2 = 2, . . . , in = n§ dž i1 i2 ⋅ ⋅ ⋅ in Ò´g,^S§¤ ± N(i1 i2 ⋅ ⋅ ⋅ in) = 0"ü½Â          a11 0 ⋅ ⋅ ⋅ 0 a21 a22 ⋅ ⋅ ⋅ 0 . . . . . . . . . . . . an1 an2 ⋅ ⋅ ⋅ ann          = ∑ i1 i2⋅⋅⋅in 12⋅⋅⋅nü (−1) N(i1 i2⋅⋅⋅in) ai11ai22 ⋅ ⋅ ⋅ ainn ¥ŒUš"‘=k ‘ a11a22 ⋅ ⋅ ⋅ ann§¤±ky(Ø" □