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1、 , , , ,313131312222222213131313333333332121212112121212323232322323232311111111323232322121212113131313313131312323232312121212333333332222222211111111 a a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a a + + + + + + += = = =
2、333333333232323231313131232323232222222221212121131313131212121211111111a a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a a例如例如( ( ( () ) ) )3232323223232323333333332222222211111111a a a aa a a aa a a aa a a aa a a a = = = =( ( ( () ) ) )3333333321212121313131312323232312121212a a a aa a
3、 a aa a a aa a a aa a a a + + + + ( ( ( () ) ) )3131313122222222323232322121212113131313a a a aa a a aa a a aa a a aa a a a + + + +33333333313131312323232321212121 13131313 33333333313131312323232321212121 12121212 33333333323232322323232322222222 11111111a a a aa a a aa a a aa a a aa a a aa a a aa
4、a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a a+ + + + = = = =一、余子式与代数余子式一、余子式与代数余子式一、余子式与代数余子式一、余子式与代数余子式在在 阶行列式中，把元素阶行列式中，把元素 所在的第所在的第 行和第行和第 列划去后，留下来的列划去后，留下来的 阶行列式叫做元素阶行列式叫做元素 的的余子式余子式，记作，记作n n n nij ij ij ija a a ai i i ij j j j 1 1 1 1 n n n nij ij ij ija a a a . . . .MMMMij ij i
5、j ij ( ( ( () ) ) )，记记ij ij ij ijj j j ji i i i ij ij ij ijMMMMA A A A+ + + + = = = =1 1 1 1叫做元素叫做元素 的的代数余子式代数余子式ij ij ij ija a a a例如例如44444444434343434242424241414141343434343333333332323232313131312424242423232323222222222121212114141414131313131212121211111111a a a aa a a aa a a aa a a aa a a aa
6、a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aD D D D= = = =44444444424242424141414134343434323232323131313114141414121212121111111123232323 a a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aMMMM= = = =( ( ( () ) ) )232323233 3 3 32 2 2 2 232323231 1 1 1MMMM
7、A A A A+ + + + = = = =. . . .23232323MMMM = = = =, , , ,44444444434343434242424241414141343434343333333332323232313131312424242423232323222222222121212114141414131313131212121211111111a a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aD D
8、D D= = = =, , , ,44444444434343434141414134343434333333333131313124242424232323232121212112121212 a a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aMMMM= = = =( ( ( () ) ) )121212122 2 2 21 1 1 1 121212121 1 1 1MMMMA A A A+ + + + = = = =. . . .12121212MMMM = = = =, , , ,3333333332323232
9、3131313123232323222222222121212113131313121212121111111144444444 a a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aMMMM= = = =( ( ( () ) ) ). . . .1 1 1 144444444444444444 4 4 44 4 4 4 44444444MMMMMMMMA A A A= = = = = = = =+ + + +. . . .个代数余子式个代数余子式对应着一个余子式和一对应着一个余子式和一行列式的每个元素分别行列式的每个元素
10、分别引理引理引理引理 一个一个一个一个 阶行列式，如果其中第阶行列式，如果其中第阶行列式，如果其中第阶行列式，如果其中第 行所有行所有行所有行所有 元素除元素除元素除元素除 外都为零，那末这行列式等于外都为零，那末这行列式等于外都为零，那末这行列式等于外都为零，那末这行列式等于 与它的与它的与它的与它的 代数余子式的乘积，即代数余子式的乘积，即代数余子式的乘积，即代数余子式的乘积，即 ij ij ij ijij ij ij ijA A A Aa a a aD D D D= = = =n n n ni i i iij ij ij ija a a aij ij ij ija a a a444444
11、444343434342424242414141413333333324242424232323232222222221212121141414141313131312121212111111110 0 0 00 0 0 00 0 0 0a a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aD D D D= = = =( ( ( () ) ) ). . . .1 1 1 1444444444242424241414141242424242222222221212121
12、141414141212121211111111333333333 3 3 33 3 3 3a a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a aa a a a+ + + + = = = =例如例如例如例如定理定理定理定理 行列式等于它的任一行（列）的各元行列式等于它的任一行（列）的各元行列式等于它的任一行（列）的各元行列式等于它的任一行（列）的各元 素与其对应的代数余子式乘积之和，即素与其对应的代数余子式乘积之和，即素与其对应的代数余子式乘积之和，即素与其对应的代数余子式乘积之和，即inininininininini i
13、 i ii i i ii i i ii i i iA A A Aa a a aA A A Aa a a aA A A Aa a a aD D D D+ + + + + + + + + += = = =2 2 2 22 2 2 21 1 1 11 1 1 1( ( ( () ) ) )n n n ni i i i, , , , , , , 2 2 2 2 , , , , 1 1 1 1= = = =证证证证nnnnnnnnn n n nn n n ninininini i i ii i i in n n na a a aa a a aa a a aa a a aa a a aa a a aa a
14、 a aa a a aa a a aD D D D2 2 2 21 1 1 12 2 2 21 1 1 11 1 1 112121212111111110 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 0+ + + + + + + + + + + + + + + + + + + + + + + + + + + += = = =二、行列式按行（列）展开法则二、行列式按行（列）展开法则二、行列式按行（列）展开法则二、行列式按行（列）展开法则nnnnnnnnn n n nn n n ni i i in n n na a a aa a a aa a a aa a a a
15、a a a aa a a aa a a a2 2 2 21 1 1 11 1 1 11 1 1 112121212111111110 0 0 00 0 0 0= = = =nnnnnnnnn n n nn n n ni i i in n n na a a aa a a aa a a aa a a aa a a aa a a aa a a a2 2 2 21 1 1 12 2 2 21 1 1 112121212111111110 0 0 00 0 0 0+ + + +nnnnnnnnn n n nn n n nininininn n n na a a aa a a aa a a aa a a aa a a aa a a aa a a a2 2 2 21 1 1 11 1 1 112121212111111110 0 0 00 0 0 0+ + + + + +
