第01讲 行列式的定义及性质

1、第1讲 1.1 n阶行列式的定义及性质第1章行 列 式1.1 n 阶行列式的定义及性质阶行列式的定义及性质二阶行列式用于解二元一次联立方程组二阶行列式用于解二元一次联立方程组求解下面的方程组)()(2122221211212111bxaxabxaxa得：,)()(122221aa212122121122211babaxaaaa)(得：,)()(211112aa121211221122211babaxaaaa)(,时当021122211aaaa,211222112121221aaaababax211222111212112aaaababax阶行列式二为称bcaddcba,211222112121

2、221aaaababax211222111212112aaaababax.2211112babaD ,2221211ababD ,22211211aaaaD 记则二元一次方程组的解为：DDxDDx2211,D称为线性方程组的系数行列式.2.2.三阶行列式用于解三元一次联立方程组三阶行列式用于解三元一次联立方程组定义三阶行列式如下：定义三阶行列式如下：333231232221131211aaaaaaaaa323122211333312321123332232211aaaaaaaaaaaaaaa332112322311312213322113312312332211aaaaaaaaaaaaaaaa

3、aa323122211211333231232221131211aaaaaaaaaaaaaaa131312121111AaAaAa3332232211aaaaM的代数余子式。为称ijijijjiijaAMA，)(aaaM3331232112aaaaM的余子式；为称ijijaM987654321例如：875439764298651)()()(3532342362484510336231)()()(987654321861942753843762951487210596844512010518045. 0,时当0333231232221131211aaaaaaaaaD33

4、32321312232221211313212111bxaxaxabxaxaxabxaxaxa对于线性代数方程组.,DDxDDxDDx332211其中，,3332323222131211aabaabaabD ,3333123221131112abaabaabaD .3323122221112113baabaabaaD 1.1.1 n 阶行列式的定义（递归法）阶行列式的定义（递归法）定义定义由n2个数aij(i,j=1,2,n)组成的n 阶行列式阶行列式当n=1时，Da11当n2时,nnAaAaAaD1112121111是一个算术式.nnnnnnaaaaaaaaaD212222111211其中其

5、中：aij称为行列式的第i行，第j列的元素;Mij是划去D的第i行第j列后的n1阶行列式;M ij 称为aij的余子式;Aij =(-1)i+j Mij称为aij的代数余子式。nnijijaaD|)det(简记为nnjnjnnnijijiinijijiinjjjiijaaaaaaaaaaaaaaaaA,)(1111111111111111111111111例例1 对角行列式、下三角行列式对角行列式、下三角行列式niiinnnnnnaaaaaaaaaa1212221112211000000000证明：证明：nnnnaaaaaa21222111000nnnnaaaaaaa3233322211000

6、nnnnaaaaaaaa434443332211000nnnnnnnnaaaaaa,1112222110.,nnnnnnaaaaa11222211000000000000121aaaaDnnn例例2 计算行列式计算行列式=(1)n(n1)/2a1a2an1an= (1)n1 an Dn-112112)2()1() 1(aaaannnn=Dn=(1)n+1 an Dn-1=(1)n1 an (1)n2 an1 Dn-2解：解：1.1.2 n 阶行列式的性质阶行列式的性质行列式的转置行列式的转置nnnnnnTaaaaaaaaaD212221212111nnnnnnaaaaaaaaaD2122221

7、11211称行列式称行列式为行列式为行列式转置行列式转置行列式性质性质1 1 行列式行列式D与其转置行列式与其转置行列式DT相等相等. .证明：证明：采用归纳法证明.下面以4阶行列式说明之。当n=1时，,1111aDaDT.TDD当n=2时，.2112221122211211aaaaaaaaD.2112221122122111aaaaaaaaDT.TDD归纳假设，对于阶数不超过3的行列式，结论成立。对于4阶行列式44434241343332312423222114131211aaaaaaaaaaaaaaaaD44342414433323134232221241312111aaaaaaaaaaa

8、aaaaaDT44342443332342322211aaaaaaaaaa44341443331342321221aaaaaaaaaa44241443231342221231aaaaaaaaaa34241433231332221241aaaaaaaaaa由归纳假设，44434234333224232211aaaaaaaaaaDT44434234333214131221aaaaaaaaaa44434224232214131231aaaaaaaaaa34333224232214131241aaaaaaaaaa44434234333224232211aaaaaaaaaaDT4443423433321

9、4131221aaaaaaaaaa44434224232214131231aaaaaaaaaa34333224232214131241aaaaaaaaaa43423332144442343213444334331221aaaaaaaaaaaaaaaa44434234333224232211aaaaaaaaaa43422322144442242213444324231231aaaaaaaaaaaaaaaa33322322143432242213343324231241aaaaaaaaaaaaaaaa43423332144442343213444334331221aaaaaaaaaaaaaaaa4

10、4434234333224232211aaaaaaaaaa43422322144442242213444324231231aaaaaaaaaaaaaaaa33322322143432242213343324231241aaaaaaaaaaaaaaaa44434234333224232211aaaaaaaaaa34332423414443242331444334332112aaaaaaaaaaaaaaaa34322422414442242231444234322113aaaaaaaaaaaaaaaa33322322414342232231434233322114aaaaaaaaaaaaaaaa对

11、二阶矩阵转置后得到：44434234333224232211aaaaaaaaaaDT34243323414424432331443443332112aaaaaaaaaaaaaaaa34243222414424422231443442322113aaaaaaaaaaaaaaaa33233222414323422231433342322114aaaaaaaaaaaaaaaa44434234333224232211aaaaaaaaaaDT44342443332341312112aaaaaaaaaa44342442322241312113aaaaaaaaaa43332342322241312114aa

12、aaaaaaaa44434234333224232211aaaaaaaaaaDT44342443332341312112aaaaaaaaaa44342442322241312113aaaaaaaaaa43332342322241312114aaaaaaaaaa44434234333224232211aaaaaaaaaaDT44434134333124232112aaaaaaaaaa44424134323124222113aaaaaaaaaa43424133323123222114aaaaaaaaaaDaaaaaaaaaaaaaaaa4443424134333231242322211413121

13、11414131312121111AaAaAaAaDT证毕阶数不超过n的行列式与其转置行列式相等,的各个代数余子式；阶行列式表示设DnAij的各个代数余子式；阶行列式表示设TijDnB则：jiijAB nnnnnnaaaaaaaaaD212222111211nnnnnnaaaaaaaaa212221212111nnBaBaBa11122111111121211111nnAaAaAa行列式也可以用其第1列的数值与其代数余子式展开。性质性质1 1 行列式行列式D与其转置行列式与其转置行列式DT相等相等. .再证明：再证明：采用归纳法证明.还以4阶行列式说明之。当n=1时，,1111aDaDT.TD

14、D当n=2时，.2112221122211211aaaaaaaaD.2112221122122111aaaaaaaaDT.TDD归纳假设，对于阶数不超过3的行列式，结论成立。对于4阶行列式44434241343332312423222114131211aaaaaaaaaaaaaaaaD44342414433323134232221241312111aaaaaaaaaaaaaaaaDT44342443332342322211aaaaaaaaaa44341443331342321221aaaaaaaaaa44241443231342221231aaaaaaaaaa34241433231332221

15、241aaaaaaaaaa43334232144434423213443443331221aaaaaaaaaaaaaaaa44434234333224232211aaaaaaaaaa43234222144424422213442443231231aaaaaaaaaaaaaaaa33233222143424322213342433231241aaaaaaaaaaaaaaaa44434234333224232211aaaaaaaaaa34243323414424432331443443332112aaaaaaaaaaaaaaaa34243222414424422231443442322131aaa

16、aaaaaaaaaaaaa33233222414323422231433342322114aaaaaaaaaaaaaaaa44434234333224232211aaaaaaaaaa44342443332341312112aaaaaaaaaa44342442322241312113aaaaaaaaaa43332342322241312114aaaaaaaaaa44434234333224232211aaaaaaaaaaDT44434134333124232112aaaaaaaaaa44424134323124222113aaaaaaaaaa43424133323123222114aaaaaaa

17、aaaDaaaaaaaaaaaaaaaa444342413433323124232221141312111414131312121111AaAaAaAaDT性质性质2 2 行列式对任一行行列式对任一行( (或列或列) )按下式展开按下式展开, ,其值相等其值相等, ,即即niAaAaAaAaDininiiiinkikik,122111njAaAaAaAaDnjnjjjjjnkkjkj,122111证明：证明：采用归纳法证明.仅就n=4，i=3时举例说明当n=2时，222221212112221122211211AaAaaaaaaaaaD归纳假设，对于阶数不超过3的行列式D，结论成立.22221

18、212AaAa对于n阶行列式设i=3,1414131312121111MaMaMaMaD44434241343332312423222114131211aaaaaaaaaaaaaaaaD按第三行展开，得将14131211MMMM,43422322344442242233444324233211aaaaaaaaaaaaaaaaD43412321344441242133444324233112aaaaaaaaaaaaaaaa42412221344441242132444224223113aaaaaaaaaaaaaaaa42412221334341232132434223223114aaaaaaaa

19、aaaaaaaa4444333332223131MaMaMaMa.4444333332223131AaAaAaAaD987654321643189731597642)()()(1268219542362. 0976431zyx643197319764zyxnnnniniinnnnniniinaaaaaaaaakaaakakakaaaaD212111211212111211nnnnininiiiinaaabababaaaaD21221111211nnnniniinnnnniniinaaabbbaaaaaaaaaaaa212111211212111211性质性质3 3（线性性质）（线性性质）利用性

20、质2即可证明。推论推论1 若行列式有一行元素全为零，则行列式的值等于零。(相当于第1式中k=0).性质性质4 若行列式有两行元素相同，则行列式的值为若行列式有两行元素相同，则行列式的值为0证明：证明：用归纳法证明：显然,行列式阶n=2 时命题 成立.设命题对 n -1 阶行列式成立，对第 i, j 行相同的 n 阶行列式D， 将第 k(ki, j)行展开,得knknkkkknlklklAaAaAaAaD22111阶行列式，列的第行是划去第其中11nlkMMAklkllkkl,)(,行相同行与第的第jiMkl),(nlMkl210 ，由归纳假设，. 0D故推论推论2 若行列式中两行元素成比例，则行列式的值为若行列式中两行元素成比例，则行列式的值为0性质性质5 将行列式的某一行乘以常数加到另一行将行列式的某一行乘以常数加到另一行(对行对行列式作倍加行变换列式作倍加行变换), 则行列式的值不变。则行列式的值不变。nnnnjnjjiniinaaaaaaaaa