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线性代数数学科学学院陆斌第一章行列式第一节二阶与三阶行列式二阶行列式三阶行列式用消元法解二元线性方程组.,22221211212111bxaxabxaxa12:122a,2212221212211abxaaxaa:212a,1222221212112abxaaxaa,得两式相减消去2x一、二阶行列式的引入;212221121122211baabxaaaa)(,得类似地,消去1x,211211221122211abbaxaaaa)(时,当021122211aaaa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)的数表)4(22211211aaaa定义)5(42221121121122211aaaaaaaa行列式,并记作)所确定的二阶称为数表(表达式即.2112221122211211aaaaaaaaD11a12a22a12a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1223D)4(3,07112121D,14121232D,21DDx11,2714DDx22.3721二、三阶行列式定义333231232221131211)5(339aaaaaaaaa列的数表行个数排成设有记,312213332112322311322113312312332211)6(aaaaaaaaaaaaaaaaaa333231232221131211aaaaaaaaa(6)式称为数表(5)所确定的三阶行列式.333231232221131211aaaaaaaaaD.列标行标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1对角线法则只适用于二阶与三阶行列式.322113aaa312312aaa312213aaa332112aaa三阶行列式的计算2.三阶行列式包括3!项,每一项都是位于不同行,不同列的三个元素的乘积,其中三项为正,三项为负.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(241124843264.14.094321112xx求解方程例3解方程左端1229184322xxxxD,652xx解得由052xx3.2xx或
本文标题:线性代数课件--1-1二阶与三阶行列式1
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