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线性代数（第五版）在以往的学习中，我们接触过二元、三元等简单的线性方程组.但是，从许多实践或理论问题里导出的线性方程组常常含有相当多的未知量，并且未知量的个数与方程的个数也不一定相等.3我们先讨论未知量的个数与方程的个数相等的特殊情形.在讨论这一类线性方程组时，我们引入行列式这个计算工具.4第一章行列式�内容提要§1二阶与三阶行列式§2全排列及其逆序数§3nnnn阶行列式的定义§4对换§5行列式的性质§6行列式按行（列）展开§7克拉默法则行列式的概念.行列式的性质及计算.————————线性方程组的求解.（选学内容）•行列式是线性代数的一种工具！•学习行列式主要就是要能计算行列式的值.§1111二阶与三阶行列式我们从最简单的二元线性方程组出发，探求其求解公式，并设法化简此公式.一、二元线性方程组与二阶行列式二元线性方程组11112211111221111122111112212112222211222221122222112222axaxbaxaxbaxaxbaxaxbaxaxbaxaxbaxaxbaxaxb+=+=+=+=⎧⎧⎧⎧⎨⎨⎨⎨+=+=+=+=⎩⎩⎩⎩由消元法，得212121211111222211111111222221212121121212122222222211111111))))((((aaaabbbbbbbbaaaaxxxxaaaaaaaaaaaaaaaa−−−−====−−−−222212121212222222221111111121212121121212122222222211111111))))((((bbbbaaaaaaaabbbbxxxxaaaaaaaaaaaaaaaa−−−−====−−−−当时，该方程组有唯一解000021212121121212122222222211111111≠≠≠≠−−−−aaaaaaaaaaaaaaaa212121211212121222222222111111112222121212122222222211111111aaaaaaaaaaaaaaaabbbbaaaaaaaabbbbxxxx−−−−−−−−====212121211212121222222222111111112121212111112222111111112222aaaaaaaaaaaaaaaaaaaabbbbbbbbaaaaxxxx−−−−−−−−====求解公式为11112211111221111122111112212112222211222221122222112222axaxbaxaxbaxaxbaxaxbaxaxbaxaxbaxaxbaxaxb+=+=+=+=⎧⎧⎧⎧⎨⎨⎨⎨+=+=+=+=⎩⎩⎩⎩122122122122122122122122111111221221112212211122122111221221112121112121112121112121222211221221112212211122122111221221baabbaabbaabbaabxxxxaaaaaaaaaaaaaaaaabbaabbaabbaabbaxxxxaaaaaaaaaaaaaaaa−−−−⎧⎧⎧⎧====⎪⎪⎪⎪−−−−⎪⎪⎪⎪⎨⎨⎨⎨−−−−⎪⎪⎪⎪====⎪⎪⎪⎪−−−−⎩⎩⎩⎩二元线性方程组请观察，此公式有何特点？�分母相同，由方程组的四个系数确定....�分子、分母都是四个数分成两对相乘再相减而得....其求解公式为11112211111221111122111112212112222211222221122222112222axaxbaxaxbaxaxbaxaxbaxaxbaxaxbaxaxbaxaxb+=+=+=+=⎧⎧⎧⎧⎨⎨⎨⎨+=+=+=+=⎩⎩⎩⎩122122122122122122122122111111221221112212211122122111221221112121112121112121112121222211221221112212211122122111221221baabbaabbaabbaabxxxxaaaaaaaaaaaaaaaaabbaabbaabbaabbaxxxxaaaaaaaaaaaaaaaa−−−−⎧⎧⎧⎧====⎪⎪⎪⎪−−−−⎪⎪⎪⎪⎨⎨⎨⎨−−−−⎪⎪⎪⎪====⎪⎪⎪⎪−−−−⎩⎩⎩⎩二元线性方程组我们引进新的符号来表示““““四个数分成两对相乘再相减””””.1112111211121112112212211122122111221221112212212122212221222122aaaaaaaaDaaaaDaaaaDaaaaDaaaaaaaaaaaa==−==−==−==−11121112111211122122212221222122aaaaaaaaaaaaaaaa记号11121112111211122122212221222122aaaaaaaaaaaaaaaa数表表达式称为由该数表所确定的二阶行列式，即11221221112212211122122111221221aaaaaaaaaaaaaaaa−−−−其中，称为元素.(1,2;1,2)(1,2;1,2)(1,2;1,2)(1,2;1,2)ijijijijaijaijaijaij========iiii为行标，表明元素位于第iiii行；jjjj为列标，表明元素位于第jjjj列.原则：横行竖列二阶行列式的计算11121112111211122122212221222122aaaaaaaaaaaaaaaa11221221112212211122122111221221aaaaaaaaaaaaaaaa=−=−=−=−主对角线副对角线即：主对角线上两元素之积－副对角线上两元素之积————————对角线法则二元线性方程组11112211111221111122111112212112222211222221122222112222axaxbaxaxbaxaxbaxaxbaxaxbaxaxbaxaxbaxaxb+=+=+=+=⎧⎧⎧⎧⎨⎨⎨⎨+=+=+=+=⎩⎩⎩⎩若令11121112111211122122212221222122aaaaaaaaDDDDaaaaaaaa====1212121211111111222222222222bbbbbbbbaaaaDDDDaaaa====1111222222221111111121212121bbbbaaaaDDDDaaaabbbb====(方程组的系数行列式)则上述二元线性方程组的解可表示为1111122122122122122122122122111111221221112212211122122111221221DDDDDDDDbaabbaabbaabbaabxxxxaaaaaaaaaaaaaaaa====−−−−====−−−−1121212112121211212121121212222211221221112212211122122111221221abbaDabbaDabbaDabbaDxxxxaaaaDaaaaDaaaaDaaaaD−−−−========−−−−例1111求解二元线性方程组⎩⎩⎩⎩⎨⎨⎨⎨⎧⎧⎧⎧====++++====−−−−1111222212121212222233332222111122221111xxxxxxxxxxxxxxxx解因为1111222222223333−−−−====DDDD00007777))))4444((((3333≠≠≠≠====−−−−−−−−====14141414))))2222((((12121212111111112222121212121111====−−−−−−−−====−−−−====DDDD21212121242424243333111122221212121233332222−−−−====−−−−========DDDD所以11111111141414142,2,2,2,7777DDDDxxxxDDDD============222222222121212133337777DDDDxxxxDDDD−−−−===−===−===−===−二、三阶行列式定义设有9999个数排成3333行3333列的数表原则：横行竖列引进记号称为三阶行列式.111213111213111213111213212223212223212223212223313233313233313233313233aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa====112233122331132132112233122331132132112233122331132132112233122331132132132231122133112332132231122133112332132231122133112332132231122133112332aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa++++++++−−−−−−−−−−−−111213111213111213111213212223212223212223212223313233313233313233313233aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa主对角线副对角线二阶行列式的对角线法则并不适用！三阶行列式的计算————————对角线法则111213111213111213111213212223212223212223212223313233313233313233313233aaaaaaaaaaaaDaaaDaaaDaaaDaaaaaaaaaaaaaaa====132132132132132132132132aaaaaaaaaaaa++++112233112233112233112233aaaaaaaaaaaa====122331122331122331122331aaaaaaaaaaaa++++132231132231132231132231aaaaaaaaaaaa−−−−122133122133122133122133aaaaaaaaaaaa−−−−112332112332112332112332aaaaaaaaaaaa−−−−注意：对角线法则只适用于二阶与三阶行列式.实线上的三个元素的乘积冠正号，虚线上的三个元素的乘积冠负号.12-412-412-412-4-221-221-221-221-34-2-34-2-34-2-34-2DDDD====例2222计算行列式解按对角线法则，有====DDDD4444))))2222(((())))4444(((())))3333((((11112222))))2222((((22221111××××−−−−××××−−−−++++−−−−××××××××++++−−−−××××××××))))3333((((2222))))4444(((())))2222(((())))2222((((2222444411111111−−−−××××××××−−−−−−−−−−−−××××−−−−××××−−−−××××××××−−−−24242424888844443232323266664444−−−−−−−−−−−−++++−−−−−−−−====....14141414−−−−====方程左端解由得2222111111111111230.230.230.230.49494949xxxxxxxx====例3333求解方程1212121222229999181818184444333322222222−−−−−−−−−−−−++++++++====xxxxxxxxxxxxxxxxDDDD,,,,666655552222++++−−−−====xxxxxxxx2222560560560560xxxxxxxx