2-1(行列式的定义)

1§2.1n阶行列式的定义§2.2行列式的性质与计算*§2.3拉普拉斯展开定理§2.4克拉默法则§2.5矩阵的秩2§2.1n阶行列式的定义二、n阶行列式的定义一、二阶行列式3一、二阶行列式存在数aaa101.1存在但方阵AOA我们知道：问题能否找到一个刻画方阵特征的数,用这个数来判别方阵何时可逆?答案可以.为了寻找刻画方阵在何时可逆的数,我们首先考虑二阶方阵的情形.422211211aaaaA设二阶方阵二阶方阵A可逆只有零解0AX.0,0222121212111xaxaxaxa即12.只有零解221a212,2xa消去0121122211xaaaa）（3.只有零解.021122211aaaa数5引入二阶行列式,2112221122211211aaaaaaaa22211211aaaaA设二阶方阵.detAA或并记作11a12a22a21a主对角线副对角线2211aa.2112aa记忆方法----对角线法则6一般地：设n阶方阵nnaaa2211nnaaa21122121nnaaa记号,detAA或并记作nnaaa2211nnaaa21122121nnaaaA称为n阶方阵A的n阶行列式,的定义，我们先给出为了进一步给出A余子式和代数余子式的定义.二、n阶行列式定义.,||nijDa7定义在n阶行列式中，把元素ija所在的第i行和第j列划去后，余下的n－1阶行列式叫做元素ija的余子式.记为ijM称ijjiijMA1为元素ija的代数余子式.例如44434241343332312423222114131211aaaaaaaaaaaaaaaaD44424134323114121123aaaaaaaaaM2332231MA.23M844434241343332312423222114131211aaaaaaaaaaaaaaaaD44434134333124232112aaaaaaaaaM1221121MA12M33323123222113121144aaaaaaaaaM444444441MMA注意行列式的每个元(位置)都分别对应着一个余子式和一个代数余子式.9观察二阶行列式计算式：2112221122211211)1(aaaaaaaa12121111AaAa||)1(|,|)1(212112221111aAaA规定一阶矩阵(a)的行列式为a.二阶行列式的值等于它的第一行的各元素与其对应的代数余子式乘积之和，这种用低阶行列式定义高一阶行列式的方法是有一般意义的.10n阶矩阵A的行列式nnnnnnaaaaaaaaaA212222111211;)det(det111111aaAn时，当）（.221112121111nnAaAaAaAn时，当）（定义上式也称为n阶行列式︱A︱按第一行的展开式.11,)det(det111111aaAn时，当）（.1112121111nnAaAaAaA特别地不能与数的绝对值混淆.333231232221131211aaaaaaaaa时，当）（32n333223221111)1(aaaaa333123213113333123212112)1()1(aaaaaaaaaa,312213332112322311322113312312332211aaaaaaaaaaaaaaaaaa12323122211211aaaaaa.312213332112322311aaaaaaaaa(1)沙路法三阶行列式计算式的记忆法322113312312332211aaaaaaaaaD333231232221131211aaaaaaaaaD13333231232221131211aaaaaaaaa332211aaa.322311aaa(2)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例求detA:273342731A解196)1214(7)94(3)218(7342)1(72332)1)(3(2734)1(1312111273342731detA1415243122421D3阶行列式计算例解按对角线法则，有D)2(2141124843264.14)3(124)2()4()2()2(2)3(2)4(16.094321112xx求解方程例解方程左端为1229184322xxxx,652xx解得由0652xx3.2xx或17行列式与矩阵的区别与联系;,1nmnnAD;,2数表数;,,3.4AAnn18例计算nnnnnaaaaaaD21222111O解nnnnnaaaaaaaD3233322211Onnnnaaaaaaaa434443332211O(下三角行列式).2211nnaaannnnnaaaaaaD21222111Onnaaa2211下三角行列式nnnnnaOaaaaaD22211211nnaaa2211上三角行列式同理1920特别地要记住!nnaaa0000002211对角行列式.,1nkknII.2211nnaaa21例计算次下三角行列式***00012aaaDnn***000)1(1211aaaaDnnnn11)1(nnnDanna1)1(111)2()1()1(Daannnn.)1(212)1(nnnaaa解])1[(212nnnDa次下三角行列式次上三角行列式同理*12aaaODnnnnnaaa212)1()1(OaaaDnn12*nnnaaa212)1()1(22230012aaan.)1(212)1(nnnaaa特别地24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例;212221121122211baabxaaaa）（，得类似地，消去1x,211211221122211abbaxaaaa）（25时，当021122211aaaa方程组的解为，211222112122211aaaabaabx）（3.211222112112112aaaaabbax.,22221211212111bxaxabxaxa若记,22211211aaaaD.,22221211212111bxaxabxaxa对于二元线性方程组系数行列式26.,22221211212111bxaxabxaxa,22211211aaaaD,2221211ababD.,22221211212111bxaxabxaxa,22211211aaaaD.2211112babaD，211222112122211aaaabaabx.211222112112112aaaaabbax,11DDx解)0.(22DDDx§2.4要介绍的Cramer法则27例.12,12232121xxxx求解二元线性方程组解1223D)4(3,07112121D,14121232D,21DDx11,2714DDx22.372128解线性方程组.0,132,22321321321xxxxxxxxx解由于方程组的系数行列式111312121D1111321211111221315,029同理可得1103111221D,51013121212D,100111122213D,5故方程组的解为:,111DDx,222DDx.133DDx.0,132,22321321321xxxxxxxxx,5111312121D1000000002012000201100020001000D计算行列式1000000002012000201100030002000)1(2012120132012201220132013)1(DD记30201220132013)1(DD记1000000002012000201100020001000D201120122013!2)1(D1002012!2011)1(220112016!20122320122013!2011)1(D3132小结1.余子式与代数余子式ija的余子式ijM行列式定义.2.1112121111nnAaAaAaAijjiijMA1代数余子式ija按第一行的展开式.33特别地nnaaa0000002211(2)对角行列式.2211nnaaannnnnaaaaaaD21222111O(1)(下三角行列式).2211nnaaa3400)4(12aaan.)1(212)1(nnnaaa.12121nnnaaa(3)次下三角行列式***00012aaaDnn3511a12a22a21a2211aa.2112aa3.对角线法则333231232221131211aaaaaaaaa332211aaa.322311aaa322113aaa312312aaa312213aaa332112aaa36作业P561.(2)(4)2.(1)