排列定义: n11a12 ain ai1ai2…·ain am为12m的全排列 其中N(i1i2……i)是全排列i1i2…in中左大右小的逆序对数目 从这个定义也可看出,一个n阶行列式中有n!项的代数(表示有正有 负)和,每一项是不在同行,不在同列的n个数之积 而且,当Ⅱ=2或n=3时,上述两个定义对应的值恰好一致。这并 不是偶然的,实际上它对任意n阶行列式都成立,我们将在以后证明 这个结论 行列式是n×n阶矩阵对应的一个数值,是关于矩阵中n2个项(可当 作n2元)一个函数
p ê EÆ£Á¤ 1ª n 1ªVg ü½Âµ a11 a12 ⋅ ⋅ ⋅ a1n a21 a22 ⋅ ⋅ ⋅ a2n . . . . . . . . . . . . an1 an2 ⋅ ⋅ ⋅ ann = ∑ i1 i2⋅⋅⋅in 12⋅⋅⋅n ü (−1) N(i1 i2⋅⋅⋅in) ai11ai22 ⋅ ⋅ ⋅ ainn Ù¥ N(i1 i2 ⋅ ⋅ ⋅ in) ´ ü i1 i2 ⋅ ⋅ ⋅ in ¥m_Séê8" lù½Âwѧ n 1ª¥k n! ê£L«kk K¤Ú§z´Ø3Ó1§Ø3Ó n êÈ" § n = 2 ½ n = 3 §þãü½ÂéATÐ"ù¿ Ø´ó,§¢Sþ§é?¿ n 1ªÑ¤á§·ò3±y² ù(Ø" 5 1ª´ n × n Ý éA꧴'uÝ ¥ n 2 £ n 2 ¤¼ê"
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
p ê EÆ£Á¤ 1ª 1ª5 1ª5 þnÝ 1ª uÙéȵ a11 a12 ⋅ ⋅ ⋅ a1n 0 a22 ⋅ ⋅ ⋅ a2n . . . . . . . . . . . . 0 0 ⋅ ⋅ ⋅ ann = a11a22 ⋅ ⋅ ⋅ ann" y²µ é ü i1 i2 ⋅ ⋅ ⋅ in§ai11ai22 ⋅ ⋅ ⋅ ainn ¥kU"U ´ in = n, in−1 = n − 1, . . . , i1 = 1§ d i1i2 ⋅ ⋅ ⋅ in Ò´g,^S§ ¤± N(i1i2 ⋅ ⋅ ⋅ in) = 0"ü½Â a11 a12 ⋅ ⋅ ⋅ a1n 0 a22 ⋅ ⋅ ⋅ a2n . . . . . . . . . . . . 0 0 ⋅ ⋅ ⋅ ann = ∑ i1 i2⋅⋅⋅in 12⋅⋅⋅n ü (−1) N(i1 i2⋅⋅⋅in) ai11ai22 ⋅ ⋅ ⋅ ainn ¥U"=k a11a22 ⋅ ⋅ ⋅ ann§¤±ky(Ø" □