第01講 行列式的定義及性質(zhì)

1、第1講 1.1 n階行列式的定義及性質(zhì)第1章行 列 式1.1 n 階行列式的定義及性質(zhì)階行列式的定義及性質(zhì)二階行列式用于解二元一次聯(lián)立方程組二階行列式用于解二元一次聯(lián)立方程組求解下面的方程組)()(2122221211212111bxaxabxaxa得：,)()(122221aa212122121122211babaxaaaa)(得：,)()(211112aa121211221122211babaxaaaa)(,時(shí)當(dāng)021122211aaaa,211222112121221aaaababax211222111212112aaaababax階行列式二為稱bcaddcba,211222112121

2、221aaaababax211222111212112aaaababax.2211112babaD ,2221211ababD ,22211211aaaaD 記則二元一次方程組的解為：DDxDDx2211,D稱為線性方程組的系數(shù)行列式.2.2.三階行列式用于解三元一次聯(lián)立方程組三階行列式用于解三元一次聯(lián)立方程組定義三階行列式如下：定義三階行列式如下：333231232221131211aaaaaaaaa323122211333312321123332232211aaaaaaaaaaaaaaa332112322311312213322113312312332211aaaaaaaaaaaaaaaa

3、aa323122211211333231232221131211aaaaaaaaaaaaaaa131312121111AaAaAa3332232211aaaaM的代數(shù)余子式。為稱ijijijjiijaAMA，)(aaaM3331232112aaaaM的余子式；為稱ijijaM987654321例如：875439764298651)()()(3532342362484510336231)()()(987654321861942753843762951487210596844512010518045. 0,時(shí)當(dāng)0333231232221131211aaaaaaaaaD33

4、32321312232221211313212111bxaxaxabxaxaxabxaxaxa對于線性代數(shù)方程組.,DDxDDxDDx332211其中，,3332323222131211aabaabaabD ,3333123221131112abaabaabaD .3323122221112113baabaabaaD 1.1.1 n 階行列式的定義（遞歸法）階行列式的定義（遞歸法）定義定義由n2個(gè)數(shù)aij(i,j=1,2,n)組成的n 階行列式階行列式當(dāng)n=1時(shí)，Da11當(dāng)n2時(shí),nnAaAaAaD1112121111是一個(gè)算術(shù)式.nnnnnnaaaaaaaaaD212222111211其中其

5、中：aij稱為行列式的第i行，第j列的元素;Mij是劃去D的第i行第j列后的n1階行列式;M ij 稱為aij的余子式;Aij =(-1)i+j Mij稱為aij的代數(shù)余子式。nnijijaaD|)det(簡記為nnjnjnnnijijiinijijiinjjjiijaaaaaaaaaaaaaaaaA,)(1111111111111111111111111例例1 對角行列式、下三角行列式對角行列式、下三角行列式niiinnnnnnaaaaaaaaaa1212221112211000000000證明：證明：nnnnaaaaaa21222111000nnnnaaaaaaa3233322211000

6、nnnnaaaaaaaa434443332211000nnnnnnnnaaaaaa,1112222110.,nnnnnnaaaaa11222211000000000000121aaaaDnnn例例2 計(jì)算行列式計(jì)算行列式=(1)n(n1)/2a1a2an1an= (1)n1 an Dn-112112)2()1() 1(aaaannnn=Dn=(1)n+1 an Dn-1=(1)n1 an (1)n2 an1 Dn-2解：解：1.1.2 n 階行列式的性質(zhì)階行列式的性質(zhì)行列式的轉(zhuǎn)置行列式的轉(zhuǎn)置nnnnnnTaaaaaaaaaD212221212111nnnnnnaaaaaaaaaD2122221

7、11211稱行列式稱行列式為行列式為行列式轉(zhuǎn)置行列式轉(zhuǎn)置行列式性質(zhì)性質(zhì)1 1 行列式行列式D與其轉(zhuǎn)置行列式與其轉(zhuǎn)置行列式DT相等相等. .證明：證明：采用歸納法證明.下面以4階行列式說明之。當(dāng)n=1時(shí)，,1111aDaDT.TDD當(dāng)n=2時(shí)，.2112221122211211aaaaaaaaD.2112221122122111aaaaaaaaDT.TDD歸納假設(shè)，對于階數(shù)不超過3的行列式，結(jié)論成立。對于4階行列式44434241343332312423222114131211aaaaaaaaaaaaaaaaD44342414433323134232221241312111aaaaaaaaaaa

8、aaaaaDT44342443332342322211aaaaaaaaaa44341443331342321221aaaaaaaaaa44241443231342221231aaaaaaaaaa34241433231332221241aaaaaaaaaa由歸納假設(shè)，44434234333224232211aaaaaaaaaaDT44434234333214131221aaaaaaaaaa44434224232214131231aaaaaaaaaa34333224232214131241aaaaaaaaaa44434234333224232211aaaaaaaaaaDT4443423433321

9、4131221aaaaaaaaaa44434224232214131231aaaaaaaaaa34333224232214131241aaaaaaaaaa43423332144442343213444334331221aaaaaaaaaaaaaaaa44434234333224232211aaaaaaaaaa43422322144442242213444324231231aaaaaaaaaaaaaaaa33322322143432242213343324231241aaaaaaaaaaaaaaaa43423332144442343213444334331221aaaaaaaaaaaaaaaa4

10、4434234333224232211aaaaaaaaaa43422322144442242213444324231231aaaaaaaaaaaaaaaa33322322143432242213343324231241aaaaaaaaaaaaaaaa44434234333224232211aaaaaaaaaa34332423414443242331444334332112aaaaaaaaaaaaaaaa34322422414442242231444234322113aaaaaaaaaaaaaaaa33322322414342232231434233322114aaaaaaaaaaaaaaaa對

11、二階矩陣轉(zhuǎn)置后得到：44434234333224232211aaaaaaaaaaDT34243323414424432331443443332112aaaaaaaaaaaaaaaa34243222414424422231443442322113aaaaaaaaaaaaaaaa33233222414323422231433342322114aaaaaaaaaaaaaaaa44434234333224232211aaaaaaaaaaDT44342443332341312112aaaaaaaaaa44342442322241312113aaaaaaaaaa43332342322241312114aa

12、aaaaaaaa44434234333224232211aaaaaaaaaaDT44342443332341312112aaaaaaaaaa44342442322241312113aaaaaaaaaa43332342322241312114aaaaaaaaaa44434234333224232211aaaaaaaaaaDT44434134333124232112aaaaaaaaaa44424134323124222113aaaaaaaaaa43424133323123222114aaaaaaaaaaDaaaaaaaaaaaaaaaa4443424134333231242322211413121

13、11414131312121111AaAaAaAaDT證畢階數(shù)不超過n的行列式與其轉(zhuǎn)置行列式相等,的各個(gè)代數(shù)余子式；階行列式表示設(shè)DnAij的各個(gè)代數(shù)余子式；階行列式表示設(shè)TijDnB則：jiijAB nnnnnnaaaaaaaaaD212222111211nnnnnnaaaaaaaaa212221212111nnBaBaBa11122111111121211111nnAaAaAa行列式也可以用其第1列的數(shù)值與其代數(shù)余子式展開。性質(zhì)性質(zhì)1 1 行列式行列式D與其轉(zhuǎn)置行列式與其轉(zhuǎn)置行列式DT相等相等. .再證明：再證明：采用歸納法證明.還以4階行列式說明之。當(dāng)n=1時(shí)，,1111aDaDT.TD

14、D當(dāng)n=2時(shí)，.2112221122211211aaaaaaaaD.2112221122122111aaaaaaaaDT.TDD歸納假設(shè)，對于階數(shù)不超過3的行列式，結(jié)論成立。對于4階行列式44434241343332312423222114131211aaaaaaaaaaaaaaaaD44342414433323134232221241312111aaaaaaaaaaaaaaaaDT44342443332342322211aaaaaaaaaa44341443331342321221aaaaaaaaaa44241443231342221231aaaaaaaaaa34241433231332221

15、241aaaaaaaaaa43334232144434423213443443331221aaaaaaaaaaaaaaaa44434234333224232211aaaaaaaaaa43234222144424422213442443231231aaaaaaaaaaaaaaaa33233222143424322213342433231241aaaaaaaaaaaaaaaa44434234333224232211aaaaaaaaaa34243323414424432331443443332112aaaaaaaaaaaaaaaa34243222414424422231443442322131aaa

16、aaaaaaaaaaaaa33233222414323422231433342322114aaaaaaaaaaaaaaaa44434234333224232211aaaaaaaaaa44342443332341312112aaaaaaaaaa44342442322241312113aaaaaaaaaa43332342322241312114aaaaaaaaaa44434234333224232211aaaaaaaaaaDT44434134333124232112aaaaaaaaaa44424134323124222113aaaaaaaaaa43424133323123222114aaaaaaa

17、aaaDaaaaaaaaaaaaaaaa444342413433323124232221141312111414131312121111AaAaAaAaDT性質(zhì)性質(zhì)2 2 行列式對任一行行列式對任一行( (或列或列) )按下式展開按下式展開, ,其值相等其值相等, ,即即niAaAaAaAaDininiiiinkikik,122111njAaAaAaAaDnjnjjjjjnkkjkj,122111證明：證明：采用歸納法證明.僅就n=4，i=3時(shí)舉例說明當(dāng)n=2時(shí)，222221212112221122211211AaAaaaaaaaaaD歸納假設(shè)，對于階數(shù)不超過3的行列式D，結(jié)論成立.22221

18、212AaAa對于n階行列式設(shè)i=3,1414131312121111MaMaMaMaD44434241343332312423222114131211aaaaaaaaaaaaaaaaD按第三行展開，得將14131211MMMM,43422322344442242233444324233211aaaaaaaaaaaaaaaaD43412321344441242133444324233112aaaaaaaaaaaaaaaa42412221344441242132444224223113aaaaaaaaaaaaaaaa42412221334341232132434223223114aaaaaaaa

19、aaaaaaaa4444333332223131MaMaMaMa.4444333332223131AaAaAaAaD987654321643189731597642)()()(1268219542362. 0976431zyx643197319764zyxnnnniniinnnnniniinaaaaaaaaakaaakakakaaaaD212111211212111211nnnnininiiiinaaabababaaaaD21221111211nnnniniinnnnniniinaaabbbaaaaaaaaaaaa212111211212111211性質(zhì)性質(zhì)3 3（線性性質(zhì)）（線性性質(zhì)）利用性

20、質(zhì)2即可證明。推論推論1 若行列式有一行元素全為零，則行列式的值等于零。(相當(dāng)于第1式中k=0).性質(zhì)性質(zhì)4 若行列式有兩行元素相同，則行列式的值為若行列式有兩行元素相同，則行列式的值為0證明：證明：用歸納法證明：顯然,行列式階n=2 時(shí)命題 成立.設(shè)命題對 n -1 階行列式成立，對第 i, j 行相同的 n 階行列式D， 將第 k(ki, j)行展開,得knknkkkknlklklAaAaAaAaD22111階行列式，列的第行是劃去第其中11nlkMMAklkllkkl,)(,行相同行與第的第jiMkl),(nlMkl210 ，由歸納假設(shè)，. 0D故推論推論2 若行列式中兩行元素成比例，則行列式的值為若行列式中兩行元素成比例，則行列式的值為0性質(zhì)性質(zhì)5 將行列式的某一行乘以常數(shù)加到另一行將行列式的某一行乘以常數(shù)加到另一行(對行對行列式作倍加行變換列式作倍加行變換), 則行列式的值不變。則行列式的值不變。nnnnjnjjiniinaaaaaaaaa