1.3行列式按行（列）展开.pptVIP免费

§1.3行列式按行(列)展开分析三阶行列式的一个规律：333231232221131211aaaaaaaaa332211aaa.322311aaa322113aaa312312aaa312213aaa332112aaa211233133222113313312311321231()()()aaaaaaaaaaaaaaa1213213233(1)aaaaa121311131112212223212223323331333132(1)(1)(1)aaaaaaaaaaaaaaa现以第二行元素为标准,将各项分组1113223133aaaaa1112233132(1)aaaaa其中来自于总结规律333231232221131211aaaaaaaaa121311131112212223212223323331333132(1)(1)(1)aaaaaaaaaaaaaaa12133233aaaa333231232221131211aaaaaaaaa11133133aaaa333231232221131211aaaaaaaaa11123132aaaa333231232221131211aaaaaaaaa1ijijijMA，称为元素的代数余子式．ija定义1余子式与代数余子式在n阶行列式中，把元素所在的第i行和第j列划去后，余下的元素按原来的顺序构成的n-1阶行列式称为元素的余子式,记作ijaija.ijM【注】行列式的每个元素分别对应着一个余子式和一个代数余子式.例如44434241343332312423222114131211aaaaaaaaaaaaaaaaD44424134323114121123aaaaaaaaaM2332231MA.23M一、行列式按某一行(列)展开n阶行列式等于它的任一行(列)的各元素与其对应的代数余子式乘积之和，即ininiiiiAaAaAaD2211ni,,2,11,2,,jn行列式按某一行(列)展开定理1122jjjjnjnjDaAaAaA或333231232221131211aaaaaaaaa121311131112212223212223323331333132(1)(1)(1)aaaaaaaaaaaaaaa212122222323aAaAaA1122()iiiiininDaAaAaA往证证明思路:②右端每一项都是D中的项,并且带有相同的符号.ijijaA①(*)式两端所含项数相同,并且各项互不相同;1111111111111111111111111jjniijijinijiijijinnnjnjnnnaaaaaaaaMaaaaaaaaD中项符号为111111iinijjijijnjaaaaa111(1(1)(1))()(1)iiniiinjjjjj1111()1(1)(1)iinjjjjji111()(1)(1)iinjjjjijijijaA中每一项可写成111111111111()111()111(1)(1)[(1)](1)(1)iiniiniiniinijijijjjjjijijjijijnjjjjjijijjijijnjaMaaaaaaaaaa0532004140013202527102135...