大学线性代数-Laplace展开定理和行列式计算方法小结

Laplace展开定理二、Laplace定理行列式按某几行或几列展开定义:12(1)kiii()22111kkjjjiiiM即，中,kn(1)个元素，按原来的顺序,余下的元素按原来的顺序,余子式.其中kiii12,,,kjjj12,,,12kjjj，11121314152122232425313233343541424344455152535455aaaaaaaaaaaaaaaDaaaaaaaaaa行）（行）（53列列42Maaaaaaaaa111315212325414345M111315aaa212325aaa414345aaa()141(1)M3234aa5254aa3524如MMDaaaaaaaaaaaaaaaaaaaaaaaaa11121314152122232425313233343541424344455152535455)5()3()1()4()3()1(M(1)aaaa22254245aaa111314aaa313334aaa515354aa4245aaaa22254245135134aa2225Laplace定理例1aabbbccccdddd12123123412340001212aabb1130abb2230abb(1)(1)(1)12121213122334cc34dd24cc24dd14cc14dd6000000000000000000000000bbbbbaaabaaaDaabb例200000000aaabbbba(1)3434()322ab(1)2323aabbabbaabba3abbaaabb00000000aaabbbbanaaabbbbabDbaa2)1()(nn例3........................kkkkrrrrkrkrccbbbcbacaaa1111111111110000()()krkr......kkkkaaaa1111......rrrrbbbb1111例4........................1111111111110000rkkrkkrrrkkraaabbbaccbcc......rrrrbbbb1111......kkkkaaaa1111()()krkr例5........................1111111111110000krkkkkkrrrrrccbbbaaaabcc()()krkr例61．利用行列式定义直接计算2．利用行列式的性质计算3．化为三角形行列式4．降阶法5．逆推公式法6．利用已知行列式（范德蒙行列式）7．加边法（升阶法）8．数学归纳法9.分拆法1．利用行列式定义直接计算001002001000000nDnn2．利用行列式的性质计算1213112232132331230000nnnnnnnaaaaaaDaaaaaa3．化为三角形行列式abbbbabbDbbabbbba4．降阶法降阶法是按某一行（或一列）展开行列式，这样可以降低一阶，更一般地是用Laplace定理00010000000000001000naaaDaa1000000000000(1)0000000001000naaaaaaaa12(1)(1)nnnnaa2nnaa.5．递推公式法1221100001000001nnnnxxDxaaaaax12321100001000001nnnxxxxaaaaax11000100(1)001nnxax1nnaxD6．利用已知行列式（Vandermonder行列式）1222211221212121122111111nnnnnnnnnnnxxxDxxxxxxxxxxxx1222212111121111()nnnnnnijnijxxxxxxxxxxx7．加边法（升阶法）加边法（又称升阶法）是在原行列式中增加一行一列，且保持原行列式不变的方法。12121212nnnnnxaaaaxaaDaaaaaxa1100nnaaD1211002,,1100100niaaaxinxx第行减第1行1211000000000njnjaaaaxxxx（箭形行列式）8．数学归纳法1221100001000001nnnnxxDxaaaaax9．拆开法(分拆法)11212212nnnnnaaaaaaDaaa1212212nnnnaaaaaaaaa2122200nnnnaaaaaa1220000nnaaa11nD1211niniia