线性代数行列式的定义

线性代数行列式的定义第一章行列式n阶行列式的定义n阶行列式的性质n阶行列式的计算用消元法解二元线性方程组.,22221211212111bxaxabxaxa12:122a,2212221212211abxaaxaa:212a,1222221212112abxaaxaa，得两式相减消去2x二三阶行列式的引入;212221121122211baabxaaaa）（，得类似地，消去1x,211211221122211abbaxaaaa）（时，当021122211aaaa方程组的解为，211222112122211aaaabaabx）（3.211222112112112aaaaabbax由方程组的四个系数确定.由四个数排成二行二列（横排称行、竖排称列）外加两竖线的数学符号定义22211211aaaa即.2112221122211211aaaaaaaaD21122211aaaa其实质是一个数：,称为二阶行列式11a12a22a12a主对角线副对角线对角线法则2211aa.2112aa二阶行列式的计算若记,22211211aaaaD.,22221211212111bxaxabxaxa对于二元线性方程组系数行列式.,22221211212111bxaxabxaxa,22211211aaaaD.,22221211212111bxaxabxaxa,2221211ababD.,22221211212111bxaxabxaxa,22211211aaaaD.,22221211212111bxaxabxaxa,2221211ababD.,22221211212111bxaxabxaxa.2211112babaD则二元线性方程组的解为,2221121122212111aaaaababDDx注意分母都为原方程组的系数行列式..2221121122111122aaaababaDDx例1.12,12232121xxxx求解二元线性方程组解1223D)4(3,07,14112121D121232D,21DDx11,2714DDx22.3721三阶行列式333231232221131211aaaaaaaaa定义322231211333233121123323322211333231232221131211aaaaaaaaaaaaaaaaaaaaaaaa称为三阶行列式.由九个数排成三行三列外加两竖线的数学符号其实质是一个数：ijjiijMA)1(333231232221131211aaaaaaaaaD在三阶行列式中划去所在的行列后，剩下的元素按原来在行列式中的位置顺序所组成的二阶行列式称为的余子式，记作。而将称为的代数余子式ijaijaijMija131312121111131312121111AaAaAaMaMaMaD322231211333233121123323322211aaaaaaaaaaaaaaaD例1计算三阶行列式121-310312D解：12312D11-30121-10310)10(3)30(1)61(2333231232221131211aaaaaaaaa332211aaa.322311aaa对角线法则注意红线上三元素的乘积冠以正号，蓝线上三元素的乘积冠以负号．说明对角线法则只适用于二阶与三阶行列式．322113aaa312312aaa312213aaa332112aaa2-43-122-4-21D计算三阶行列式例２解按对角线法则，有D4)2()4()3(12)2(21)3(2)4()2()2(241124843264.14n阶行列式阶行列式称为个数，符号为设naaaaaaaaaDnnjiannnnnnnij2122221112112...1,,)1(1111111212111111njaAnAaAaAanaDjjnnn的代数余子式是其中其值余子式在Dn中划去元素aij所在的行和列,剩下的n-1阶行列式称为元素aij的余子式,记为MijnnnjnjnnijijiinijijaiinjjijaaaaaaaaaaaaaaaaM111,11,11,11,1,11,11,1,11111111即ijjiijMA)1(代数余子式。阶行列式例：计算nnaaan...2211。解：nnnnnnnnnaaaaaaaaaaaaaD...||............221122113322112211(对角行列式)。阶行列式例：计算nnnnaaaaaan............21222111。解：nnnnnnnnnnnnnaaaaaaaaaaaaaaaaD...||........................22111,12211333221122211(下三角行列式)