行列式高等代数

1补充：行列式的定义行列式的性质方程组的解与行列式的关系2111122111222112112221221122221122211212221211122()01axaxbaaaaxbabaaxaxbaaaababaxaa例：设有二元一次线性方程组，当时，二阶行列式行列式的定义11112112211221112211221121212222111122112211222122112212221,ababxaaaaaaabbaaaaaababaabbaaaaaabb定义则：二阶行列式：311112212112222121122211211111211122122212222,axaxbaxaxbaaaaxxaaaaaaaabbbb这时，方程组的解为用行列式表示：412121223132013021(2)0322,232(2)3313133221302031323131332xxxxxx例：解方程组的解解：利用行列式表示52111212122212::.由个数排成行列的表格，两边以竖线，成为一个阶行列式元素阶行列的定义式：nnnnnnijnnnnaaaaaaaaaan6111212122212:nnijnnijnnjiaaaaaaaaiaA=jMMaijijijijnaa余子式的去掉第行和第列后的行列式的:代数余子式(-1)值阶行列式的定义：nnnjnjnnijijiinijijiinjjjiijaaaaaaaaaaaaaaaaA111111111111111111111111)1(8111111121212221111121211121111det()2=对于行列式对于(）定义为：定义1阶阶行列式值第一上式称为行列按的展开式。行式nnnnnnnnnjjjaanaaaaaaaAaAaAaaaaAn111213212223313233222321232122111213323331311223333132112233233212213323311223311321321123321221313213222313133()()()阶行列式aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa2231aa101111121213132213054112213054110535302(1)(2)(1)(1)(1)1-14-14120(1)51(2)3(1)54(1)31042(5)(2)(23)(1)359:111213例解计算aAaAaA1111223312233113211111221331211222233231132233331213222321123321221331322311231332331112132122230设有三元一次线性方程组当时，方程组有惟一解：aaaaaaaaaaxaxaxbaxaxaxbaxaaaaxaxbaaaaaaaaaaaaaaaabbaabxa11131112212321223133313211121311121321222321222331323331321122333313233323,,aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaabbbbbbaxx122行列式的性质性质1n阶行列式可按任意行或列的展开式来计算.按第r行的展开式为：11221阶行列式值rrrrrrrnnnjrjjnaAaAaAaA按第s列的展开式为11221阶行列式值nnsssssssnsiiinaAaAaAaA13性质2nnnnnnaaaaaaaaa212222111211.212221212111nnnnnnaaaaaaaaa行列式转置，行列式值不变.即:141111111111.(1)jnjnnnjnnnnjnnaaaaaaakkkaaaaa某一行（列）的公因子可以提出如性:质3(2)（行列式的“加法”）nnnnsnsnssssnaaacbcbcbaaa21221111211nnnnsnssnnnnnsnssnaaacccaaaaaabbbaaa212111211212111211（注意：只拆一行，其余行不变）性质4:任意对换行列式的两行（或两列）元素，其值变号.如2311315211321132.31522311121311121311)(31rr(1)()(2)相同行列式：行列式有两行列,则行列式值=0成比例有两行(列)，行列则式值=0推论k5.把行列式的某行（列）的倍加到另一行（列）上，行列式的性质值不变18111111111111111ijnijjnttitjtnttitjtjtnnninjnnnninjnjnnaaaaaakaaaaaaaaakaaaaaaaaakaaajki列乘上，加入到第列11112211211222221122nnnnnnnnnnaxaxaxbaxaxaxbaxaxaxb3.1克莱姆法则（）定理当系数行列式3.方程组的解与行列式的关系不为零时，方程组有惟一解：111,111,11212,122,121,1,1jjjjjjjnjjnnnnnnnaaaaaaaaaaaaaaa111,11,11212,12,121,1,1111,111,11212,122,1212,jjjjjjjjjjjjnnnnnnnjnnnaaaaaaaaaaaaaaaaaaaaabbbax1,1,1(1)jnjjnnnnnaaaaajn111122121122221122000nnnnnnnnnaxaxaxaxaxaxaxaxax：如下结论成立：3.2对于齐次线性方程组1112121222120nnnnnnaaaaaaaaa方程组只有零解.1112121222120nnnnnnaaaaaaaaa方程组有非零解.