行列式化簡計算技巧竅門實題

.\行列式化簡計算技巧和實題操練——Zachary.技巧:技巧1:行列式與它的轉置行列式的值相等,即D=DT謝謝閱讀aaKaaaKa1112L1n1121Kn1aaaaaa21222n1222n2MMMMMMaaKaaaKan1n2nn1n2nnn技巧2:互換行列式的任意兩行(列),行列式的值將改變正負號謝謝閱讀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記號的外面謝謝閱讀babaKbaaaKa111112K11n1112K1nbababaaaanb22122222n21222nMMMMMMni1babaKbaaaKann1nn2nnnn1n2nn技巧4:行列式具有分行(列)相加性aaKaaaKaaaKa11121n11121n11121nMMMMMMMMMbcbcKbcbbKbccKct1t1t2t2tntnt1t2tnt1t2tnMMMMMMMMMaaKaaaKaaaKan1n2nnn1n2nnn1n2nn技巧5:將行列式的某一行(列)的各元素乘以同一數k后加到另一行(列)對應的元素上,行列式的值不變精品文檔放心下載.\aaKaaaKa11121n11121nMMMMMMaaKaakaakaKakas1s2sns1t1s2t2sntnMMMMMMaaKaaaKat1t2tnt1t2tnMMMMMMaaKaaaKan1n2nnn1n2nn技巧6:分塊行列式的值等于其主對角線上兩個子塊行列式的值的乘積感謝閱讀aKa0K0111mMMMMaKabKbaKa0K0111m111nm1mm111m111naKabKbMMMMm1mmn1nncKcbKbn1nmn1nn技巧7:[拉普拉斯按一行(列)展開定理]行列式等于它的任一行(列)的各元素與其對應的代數余子式乘積之和感謝閱讀nA(i1,2,LnA(j1,2,L,n)Da,n)aikikkjkjk1k1二.解題方法:方法1:對于2階行列式和3階行列式,可以直接使用對角線法則進行計算感謝閱讀aaaaaa,1112aa112212212122aaa111213aaaaaaaaaaaaaaaaaaaaa212223112233122331132132112332122133132231aaa313233.\方法2:上三角行列式,下三角行列式,主對角線行列式,副對角線行列式感謝閱讀aaKaa0K01112L1n11L0aanaa0nM222na,2122Ma,MMiiMMii00Ki1Ki1aaaannn1n2nn12Nn

O2

L(其余未寫出元素均為零),1 2 nn1(1)n(n21)L(其余未寫出元素均為零)感謝閱讀1 2 n方法3:若行列式中有兩行(列)對應元素相等,則此行列式的值等于零感謝閱讀aaeibbfjccgkddhl

0方法4:若行列式中有一行(列)的元素全為零,則此行列式的值為零謝謝閱讀0 a e i0 b f j0 c g k0 d h l

0方法5:若行列式中有兩行(列)元素成比例,則此行列式的值等于零精品文檔放心下載akaeibkbfjckcgkdkdhl

0.\實題操練：計算下列行列式的值:習題1:1 2 01 1 43 1 8解答:1 2 01141182(4)30(1)(1)0132(1)81(4)(1)4318感謝閱讀習題3:12345678910111213141516解答:123411315678cc5171214313141516131151習題4:333x333x333x333x3333解答:.\333x3x33x333x33c4cx3x333x3331i2ixx333x3333333x133x3rr133x3x13x332100xx1x333rrx0x0x311333rr000x41133x3rrx0x0xx42300xx000x習題5:abababab11121314abababab12222324abababab13233334abababab14243444解答:.\abababab11121314按第一列展開,abababab12222324abababab13233334abababab14243444abababababababababababab222324121314121314121314abab ab ababab ab ababab ab ababab ab ab謝謝閱讀11233334122333341322232414222324abababababababababababab243444243444243444233334abababababab121314121314有兩行元素成比例,因此值為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習題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:.\將行列式上下翻轉后再左右翻轉,不難得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習題7:x10K000x1K0000xK00MMMMM000Kx1aaaKaxann1n221解答:按第一列展開,得D的遞推公式nK1000x1K00DxDa(1)n10xK00nn1nMMMM00Kx1DxDann1nDxDan1n2n1DxDan2n3n2MDxDa212xa1 1將上述各式的兩邊分別乘以1,x,x2,L,xn1后全部相加,并化簡,得:感謝閱讀Dxnaxn1axn2La xa精品文檔放心下載n 1 2 n1 n.\習題8:a bO Na bc d

(其余未寫出元素均為