行列式的性质及展开计算

11a12a22a21a对角线法则2211aa.2112aa二阶行列式的计算二元线性方程组,2221121122212111aaaaababDDx.2221121122111122aaaababaDDx.,22221211212111bxaxabxaxa的解为333231232221131211aaaaaaaaa332211aaa.322311aaa三阶行列式的计算-------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．322113aaa312312aaa312213aaa332112aaa11a111213212223313233aaaaaaaaa21a31a12a22a32an阶行列式的定义12121212111212122212(1).nnniiiiiniiiinnnnnnaaaaaaaaaDaaa2由n个数组成的n阶行列式等于所有取自不同行不同列的n个元素的乘积的代数和记作定义).det(ija简记作的元素．称为行列式数)det(ijijaa上三角行列式12111222000nnnnaaaaaa1122.nnaaa下三角行列式11222112000nnnnaaaaaa1122.nnaaa12.n12n对角行列式特别的，第二节行列式的性质一、行列式的性质二、应用举例三、小结一、行列式的性质性质1行列式与它的转置行列式相等.行列式称为行列式的转置行列式.TDD记nnaaa2211nnaaa21122121nnaaaD2121nnaaannaaa2112TDnnaaa2211改变书写方向121321112333332221aDaaaaaaaa213121321122333132TaDaaaaaaaa112233122331132132aaaaaaaaa132231122133112332.aaaaaaaaa112233132132122331aaaaaaaaa132231112332122133.aaaaaaaaa112131122232132333aaaaaaaaa111213aaa212223aaa证明：nnnnnnaaaaaaaaaD212222111211nnnnnnTbbbbbbbbbD212222111211设则jiijab),,2,1,(nji由行列式定义nnnjjjnjjjjjjTbbbD21212121)()1(Daaannnjjjnjjjjjj21212121)()1(说明：行列式中行与列地位相同，对行成立的性质对列也成立，反之亦然。性质2互换行列式的两行（列）,行列式变号.111213212223313233aaaDaaaaaa1112131313233212223aaaDaaaaaa交换2，3行111213212223313233aaaDaaaaaa112233122331132132aaaaaaaaa132231122133112332.aaaaaaaaa111213212223313233aaaDaaaaaa112233122331132132aaaaaaaaa132231122133112332.aaaaaaaaa1112131313233212223aaaDaaaaaa111213313233212223aaaaaaaaa113121aaa123222aaa113223123321133122aaaaaaaaa133221113322123123aaaaaaaaaD111213212223313233aaaDaaaaaa112233122331132132aaaaaaaaa性质2互换行列式的两行（列）,行列式变号.证明：设nnnntnttsnssnaaaaaaaaaaaaD21212111211交换s、t两行，得nnnnsnsstnttnaaaaaaaaaaaaD212121112111s行t行由行列式定义可知，D中任一项可以写成ntsntsnjtjsjjjjjjaaaa111)()1(因为nstntsnjsjtjjnjtjsjjaaaaaaaa1111（2）（1）显然这是1D中取自不同行、不同列的n个元素的乘积，而且（2）式右端的n个元素是按它们在1D中所处的行标为自然顺序排好的。因此nstnstnjsjtjjjjjjaaaa111)()1(是1D中的一项。（3）因为，排列ntsjjjj1与排列nstjjjj1的奇偶性相反，所以项（1）与项（3）相差一符号，这就证明了D的任一项的反号是1D中的项，同样可以证明1D中的任一项的反号也是D中的项。因此，D＝－D1记法行列式的第s行：sr行列式的第s列：sc交换s、t两行：tsrr交换s、t两列：tscc例如推论如果行列式有两行（列）完全相同，则此行列式为零.证明互换相同的两行，有.0D,DD,571571266853.825825361567567361266853性质3行列式的某一行（列）中所有的元素都乘以同一数，等于用数乘此行列式.kk111221121iiinnnnnnaaaakakakaaa121112112iiinnnnnnaaaaaaaaak推论行列式的某一行（列）中所有元素的公因子可以提到行列式符号的外面．性质４行列式中如果有两行（列）元素成比例，则此行列式为零．证明12121112112iiinnnnnniiinaaakaaaaaaakaka12111211212nnnnniiiniiinaaaaaaaaaaaak.0Ex.1设.33323123222113121153531026aaaaaaaaa求解利用行列式性质,有11121321222331323335621350aaaaaaaaa333231232221131211aaaaaaaaa,115)3(2.301112132122233132335532335aaaaaaaaa333231232221131211aaaaaaaaa352()性质5若行列式的某一列（行）的元素都是两数之和.(34)(52)612763851(486)9542D则D等于下列两个行列式之和：例如12712763863859559542634526142684D性质5动画演示例.127051127051131222201131222175201131)2(2)2(175性质６把行列式的某一列（行）的各元素乘以同一数然后加到另一列(行)对应的元素上去，行列式不变．njnjninjjinjiaaaaaaaaaaaa12222111111njnjnjninjjjinjjijiaakaaaaakaaaaakaaakrr)()()(1222221111111k例如性质６把行列式的某一列（行）的各元素乘以同一数然后加到另一列(行)对应的元素上去，行列式不变．例如3211128211D212cc7211120118例１二、应用举例计算行列式常用方法：利用运算把行列式化为上三角形行列式，从而算得行列式的值．jikrr3111131111311113D3111131111311113D6666131111311113111113116113111131111020060020000248.解1234rrrr16r21rr31rr41rr例2计算行列式abbbbabbDbbabbbba解3333abbbbababbabbababbbaD将第2,3,4列都加到第1列得11311bbbabbabbabbba10003000000bbbabababab33().abab例3计算阶行列式nabbbbabbbbabbbbaD解abbbnababbnabbabnabbbbna1111D将第都加到第一列得n,,3,2abbbabbbabbbbna1111)1(babababbbbna1)1(00.)()1(1nbabna(行列式中行与列具有同等的地位,行列式的性质凡是对行成立的对列也同样成立).计算行列式常用方法：(1)利用定义;(2)利用性质把行列式化为上三角形行列式，从而算得行列式的值．三、小结行列式的性质1234234113412412301111011211011110Ex1.123413411014121123160123401131002220111123423411341241231234011310004400041111101131101111011110100300100001011110112110111103111301131013110311111011311011110311130113101311011110100300100001111110113110111103111301131013110111101003001000011111101131101111031113011310131102141312125232702510212110112030321Ex2.122222224223222243521110531313241310212110112030321102101320222032126132204107513222232113212111321214131212523270258156215072512411321253207251241056201500725(3)3521110513132413D解：3521110513132413D045100412131302134510412213251022121130342018113404222232222222221242321rrrrrr2000010022220001(4)解:12r2r2000010022200001=-4.思考题：35211105,13132413D设求11121314AAAA解：11121314AAAA11051313241311114例nD001030100211111箭形行列式目标：把第一列化为0011a成三角形行列式...nccccn12311123nini21111100200030000)11(!2niinEx.4321xaaaaxaaaaxaaaaxD)4,3,2,1,(iaxi（可以化为箭形行列式）14131312rrrrrrrraxxaaaxxaaxxaaaax413121100000))(())((4321axaxaxax10010101001143211axaaxaaxaaxx4321cccc41)(iiax1000010000104324211axaaxaaxaaxaaxxii