[线性代数]第一章行列式1 二阶与三阶行列式的引入2n阶行列式的定义3行列式的性质4余子式与代数余子式5行列式的展开定理6线性方程组的Gramer 法则7典型例题回顾用消元法解二元线性方程组:⎩⎨⎧=+=+,,22221211212111b x a x a b x a x a :2x 消去◇二阶行列式2122211211b b a a a a ,)(212221*********b a a b x a a a a -=-:1x 消去,)(211211*********a b b a x a a a a -=-时，当021122211≠-a a a a ，211222112122211a a a a b a a b x --=.211222112112112a a a a a b b a x --=1.二阶与三阶行列式的引入22211211a a a a [定义1]22211211a a a a 21122211a a a a -=:记号主对角线副对角线[二阶行列式计算:对角线法则]2211a a =2112a a -22211211a a a a :224列的数表行个数排成设有，211222112122211a a a a b a a b x --=.211222112112112a a a a a b b a x --=21122211a a a a -代数式称为该数表所确定的二阶行列式.,22211211a a a a D =⎩⎨⎧=+=+.,22221211212111b x a x a b x a x a 二元方程组:[系数行列式][二元方程组的Gramer 法则]⎩⎨⎧=+=+.,22221211212111b x a x a b x a x a 22211211a a a a D =,2221211a b a b D =⇒211222112122211a a a a b a a b x --=DD x 11=⎩⎨⎧=+=+.,22221211212111b x a x a b x a x a 22211211a a a a D =.2211112b a b a D =⇒.211222112112112a a a a a b b a x --=⎩⎨⎧=+=+.,22221211212111b x a x a b x a x a 22211211a a a a D =,2221211a b a b D =⇒DD x 11=DD x 22=,2DD =,1D D =211222112122211a a a a b a a b x --=211222112112112a a a a a b b a x --=⎪⎩⎪⎨⎧,021122211时当≠-=a a a a D :方程组有唯一解⎩⎨⎧=+=+.,22221211212111b x a x a b x a x a 二元方程组:,,22211211a a a a D =其中,2221211a b a b D =.2211112b a b a D =,,2211D D x D D x ==,021122211时当≠-=a a a a D :方程组有唯一解⎩⎨⎧=+=+.,22221211212111b x a x a b x a x a 二元方程组:,,22211211a a a a D =其中,2221211a b a b D =.2211112b a b a D =,2,1,==i D D x ii ,021122211时当≠-=a a a a D :方程组有唯一解⎩⎨⎧=+=+.,22221211212111b x a x a b x a x a 二元方程组:,,22211211a a a a D =其中,2221211a b a b D =.2211112b a b a D =⎩⎨⎧=+=-.12,12232121x x x x 解1223-=D ,07≠=112121-=D ,14=121232=D ,21-=D D x 11=,2714==D D x 22=.3721-=-=于是[例1]:求解二元线性方程组◇三阶行列式⎪⎩⎪⎨⎧=++=++=++333323213123232221211313212111b x a x a x a b x a x a x a b x a x a x a 321333231232221131211b b b a a a a a a a a a 用消元法解二元线性方程组:通过两次消元可解得:,11DD x ==D 11a 32a 2312a a +21a 13a +3322a a 31a 21a 33a 12a -3223a a -2213a a -11a 31a =1D 1b 32a 2312a a +2b 13a +3322a a 3b 33a 12a -3223a a -2213a a -1b 2b 3b[定义2]333231232221131211a a a a a a a a a 312213332112322311312312322113332211a a a a a a a a a a a a a a a a a a ---++=333231232221131211a a a a a a a a a 称为上述数表所确定的三阶行列式.:339列的数表行个数排成设有代数式=D 11a 32a 2312a a +21a 13a +3322a a 31a 21a 33a 12a -3223a a -2213a a -11a 31a[三阶行列式的计算: 对角线法则]特别注意对角线法则只适用于二阶与三阶行列式．332211a a a =.322311a a a -322113a a a +312312a a a +312213a a a -332112a a a -333231232221131211a a a a a a a a a [例]243122421----=D )3(12-⨯⨯+.14-=)2(21-⨯⨯=)4(4)2(-⨯⨯-+)3(2)4(-⨯⨯--)2()2(2-⨯-⨯-411⨯⨯-[三元方程组的Gramer 法则],0333231232221131211≠=a a a a a a a a a D 则三元线性方程组的解为:,)3,2,1(==i DD x ii ,3332323222131211a a b a a b a a b D =如果三元方程组⎪⎩⎪⎨⎧=++=++=++333323213123232221211313212111,,b x a x a x a b x a x a x a b x a x a x a 的系数行列式,3333123221131112a b a a b a a b a D =.3323122221112113b a a b a a b a a D =其中解则设,)(2c bx ax x f ++=,0)1(=++=c b a f ,324)2(=++=c b a f ,2839)3(=+-=-c b a f .28)3(,3)2(,0)1(=-==f f f 使求一个二次多项式),(x f [例2]⎪⎩⎪⎨⎧,20139124111-=-=D ,4013281231101-=-=D ,2028393240113-=-=D ,6012891341012==D,20139124111-=-=D ,4013281231101-=-=D ,2028393240113-=-=D ,6012891341012==D 由Gramer 法则得,21==DD a ,32-==D D b ,13==D D c 故所求多项式为.132)(2+-+=x x x f我们的问题元方程组对于n ⎪⎩⎪⎨⎧=+++=+++=+++,,,22112222212111212111n n nn n n n n n n b x a x a x a b x a x a x a b x a x a x a (1) 是否也有如此结构简洁的求解公式？是什么？如有i ii D D n i DD x ,,),,2,1()2( ==线性代数严钦容的线性代数课件仅用于网络课堂例排列32514 中，[定义1]n 个不同的自然数, 规定由小到大为标准次序.◇排列的逆序数3 2 5 1 4逆序逆序逆序则称这若中在一个排列,,)(21s t n s t i i i i i i i 两个数构成一个.逆序2. n 阶行列式的定义[定义2]一个排列中逆序的总数称为此排列的逆序数.[例]排列32514 的逆序数.3 2 5 1 4122:)(21的逆序数记为排列n j j j )32514(τ）（n j j j 21τ逆序数为奇数的排列称为奇排列, 逆序数为偶数的排列称为偶排列..500212=++++=奇排列333231232221131211a a a a a a a a a D =322113312312332211a a a a a a a a a ++=332112322311312213a a a a a a a a a ---◇三阶行列式展开式的规律,321321j j j a a a 213132321321j j j 312231123321j j j +符号：-符号：展开式的一般项为个全排列：的项的列标63,2,1)(6321正好取遍j j j )(321j j j τ)(321j j j τ022311偶排列奇排列∑=)(321333231232221131211321321j j j j j j a a a a a a a a a a a a )(321)1(j j j τ-,321321j j j a a a 213132321321j j j 312231123321j j j +符号：-符号：展开式的一般项为个全排列：的项的列标63,2,1)(6321正好取遍j j j )(321j j j τ)(321j j j τ022311偶排列奇排列不同行阶行列式等于所有取自个数组成的由n n 2[定义3]).det(ij a 简记为元．的称为行列式数),(j i D a ij .)1()(21)(12121∑-n nn j j nj j j j j j a a a τnnn n n n a a a a a a a a a D 212222111211=记作◇n 阶行列式的定义特别：一阶行列式aa =:个元素的乘积的代数和不同列的n[关于定义的说明]○行列式是一种特定的算式,代表一个数○n阶行列式是n!项的代数和○每项都是位于不同行不同列n个元素的乘积∑-=)(21)(12121)1(nnniiniiiii i aaaDτ○行列式的等价定义:∑-=)(21)(12121)1(nnnjjnjjjjjj aaaDτ对比[例1]计算上三角行列式:.00022211211nnnn a a a a a a D=,,2121中在于是nnj j j a a a ,0,才可能不为时当ij a i j ≥解,0,,2,121时才可能不为仅当n j j j n ≥≥≥ ,从而仅当()nn n a a a D 221112)1(τ-=.2211nn a a a =由定义,0,,2,1212121才可能不为时n nj j j n a a a n j j j ===nnλλλλλλ2121=对角行列式：nn nnnn a a a a a a a a a221122211211000=nnnnn n a a a a a a a a a 221121222111000[例2]计算行列式:4003002001000=D .43214321j j j j a a a a D 的展开式的一般项为则,0,4111=≠j a j 时当,41只能等于所以j ,1,2,3432===j j j 解,41322314a a a a 只有即行列式中不为零的项()4321)1(4321⋅⋅⋅-=τD .24=从而,0否则该项为,)(44⨯=ij a D 记同理可得()nn n nλλλλλλ212121)1(--=[公式: 反对角行列式]解的有两项：含有3x 44332211)1234()1(a a a a τ-,221)1(3x x x x -=⋅⋅⋅-=.13-的系数为故x ()433422111243)1(a a a a τ-,13x x x x =⋅⋅⋅=[例1]求多项式1211123111211)(x x x x x f -=.3的系数中x线性代数严钦容的线性代数课件仅用于网络课堂。