行列式化简计算技巧窍门实题

.\行列式化简计算技巧和实题操练——Zachary.技巧:技巧1:行列式与它的转置行列式的值相等,即D=DT谢谢阅读aaKaaaKa1112L1n1121Kn1aaaaaa21222n1222n2MMMMMMaaKaaaKan1n2nn1n2nnn技巧2:互换行列式的任意两行(列),行列式的值将改变正负号谢谢阅读aaKaaaKa1112L1n2122L2naaaaaa21222n11121nMMMMMMaaKaaaKan1n2nnn1n2nn技巧3:行列式中某一行(列)的所有元素的公因子可以提到行列式记号的外面谢谢阅读babaKbaaaKa111112K11n1112K1nbababaaaanb22122222n21222nMMMMMMni1babaKbaaaKann1nn2nnnn1n2nn技巧4:行列式具有分行(列)相加性aaKaaaKaaaKa11121n11121n11121nMMMMMMMMMbcbcKbcbbKbccKct1t1t2t2tntnt1t2tnt1t2tnMMMMMMMMMaaKaaaKaaaKan1n2nnn1n2nnn1n2nn技巧5:将行列式的某一行(列)的各元素乘以同一数k后加到另一行(列)对应的元素上,行列式的值不变精品文档放心下载.\aaKaaaKa11121n11121nMMMMMMaaKaakaakaKakas1s2sns1t1s2t2sntnMMMMMMaaKaaaKat1t2tnt1t2tnMMMMMMaaKaaaKan1n2nnn1n2nn技巧6:分块行列式的值等于其主对角线上两个子块行列式的值的乘积感谢阅读aKa0K0111mMMMMaKabKbaKa0K0111m111nm1mm111m111naKabKbMMMMm1mmn1nncKcbKbn1nmn1nn技巧7:[拉普拉斯按一行(列)展开定理]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感谢阅读nA(i1,2,LnA(j1,2,L,n)Da,n)aikikkjkjk1k1二.解题方法:方法1:对于2阶行列式和3阶行列式,可以直接使用对角线法则进行计算感谢阅读aaaaaa,1112aa112212212122aaa111213aaaaaaaaaaaaaaaaaaaaa212223112233122331132132112332122133132231aaa313233.\方法2:上三角行列式,下三角行列式,主对角线行列式,副对角线行列式感谢阅读aaKaa0K01112L1n11L0aanaa0nM222na,2122Ma,MMiiMMii00Ki1Ki1aaaannn1n2nn12Nn

O2

L(其余未写出元素均为零),1 2 nn1(1)n(n21)L(其余未写出元素均为零)感谢阅读1 2 n方法3:若行列式中有两行(列)对应元素相等,则此行列式的值等于零感谢阅读aaeibbfjccgkddhl

0方法4:若行列式中有一行(列)的元素全为零,则此行列式的值为零谢谢阅读0 a e i0 b f j0 c g k0 d h l

0方法5:若行列式中有两行(列)元素成比例,则此行列式的值等于零精品文档放心下载akaeibkbfjckcgkdkdhl

0.\实题操练：计算下列行列式的值:习题1:1 2 01 1 43 1 8解答:1 2 01141182(4)30(1)(1)0132(1)81(4)(1)4318感谢阅读习题3:12345678910111213141516解答:123411315678cc5171214313141516131151习题4:333x333x333x333x3333解答:.\333x3x33x333x33c4cx3x333x3331i2ixx333x3333333x133x3rr133x3x13x332100xx1x333rrx0x0x311333rr000x41133x3rrx0x0xx42300xx000x习题5:abababab11121314abababab12222324abababab13233334abababab14243444解答:.\abababab11121314按第一列展开,abababab12222324abababab13233334abababab14243444abababababababababababab222324121314121314121314abab ab ababab ab ababab ab ababab ab ab谢谢阅读11233334122333341322232414222324abababababababababababab243444243444243444233334abababababab121314121314有两行元素成比例,因此值为0,由于行列式ababab和ababab222324222324abababababab243444233334bbbbbbbbb234234234Daabbabababa2bbababab(aabba2bb)ababab12142333341242333341214124233334aaaaaaaaa234234234(aabba2bb)(aabbaabbaab2a2bb)121412434232334243324ab(abab)[ab(abab)ab(abab)]142112433223342332ab(abab)(abab)(abab)14211232234334ab(abab)(abab)(abab)14122123323443ab3(abab)14ii1i1i1习题6:a4(a1)4(a2)4(a3)4(a4)4a3(a1)3(a2)3(a3)3(a4)3a2(a1)2(a2)2(a3)2(a4)2aa1a2a3a411111解答:.\将行列式上下翻转后再左右翻转,不难得11111aa1a2a3a4D(1)4321a2(a1)2(a2)2(a3)2(a4)25(a1)3(a2)3(a3)3(a4)3a3a4(a1)4(a2)4(a3)4(a4)411111a4a3a2a1a(1)4321(1)4321(a4)2(a3)2(a2)2(a1)2a2(a4)3(a3)3(a2)3(a1)3a3(a4)4(a3)4(a2)4(a1)4a44!3!2!1!288习题7:x10K000x1K0000xK00MMMMM000Kx1aaaKaxann1n221解答:按第一列展开,得D的递推公式nK1000x1K00DxDa(1)n10xK00nn1nMMMM00Kx1DxDann1nDxDan1n2n1DxDan2n3n2MDxDa212xa1 1将上述各式的两边分别乘以1,x,x2,L,xn1后全部相加,并化简,得:感谢阅读Dxnaxn1axn2La xa精品文档放心下载n 1 2 n1 n.\习题8:a bO Na bc d

(其余未写出元素均为