线性代数英文课件：ch1_2 Properties of Determinants

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1、Math. Dept., Wuhan University of Technology Sec.2 Properties of DeterminantsThese properties need to be memorized.性质的推导不重要，性质要记住，要会应用性质的推导不重要，性质要记住，要会应用Math. Dept., Wuhan University of TechnologyDT (or ) D is the transposed D. 112111222211112121222122nnnnnnnTnnnnnaaaaaaDaaaaaaaaaaDaa Transposed dete

2、rminants (转置行列式）转置行列式）Math. Dept., Wuhan University of Technology【Property 1】 D =DT Proof Let 111212122212nnTnnnnbbbbbbDbbb That is,( ,1,2,)jiiji jnba 1 212()12( 1)nnj jjTjjnjDb bb 1 212()12( 1)nnj jjjjj naaa D Math. Dept., Wuhan University of Technology【Property 2】If two rows or two columns of a de

3、terminant are interchanged, then the determinant changes by a sign. ProofLet 1112121222112nnnnnnbbbbbbDbbb interchanging row k and row s, that is, when ik, s, ,ijijba ,kjsjsjkjbaba 11()11( 1)ksnksnjjjjjk jsjnjDbbbb and when i=k, s, . is obtained by Math. Dept., Wuhan University of Technology11()1( 1

4、)sknsknjjjjjk jsjnjaaaa D Let ir denote row i of a determinant,ic denote column i of a determinant,ijrrInterchanging row i and row j, marked as ijcc Interchanging column i and column j, marked as 11()1( 1)ksnksnjjjjjs jkjnjaaaa Math. Dept., Wuhan University of Technology11212332112113231231213212121

5、31ccrr+r3c1r1c2fromtofromtoE.g.,【 Corollary 1 】 if two rows or two columns of a determinant are identical, the determinant is equal to 0.246314314 23rr246314314 0DDD Math. Dept., Wuhan University of Technology【Property 3】 The common factor（公因子）（公因子） for all the elements of one row or one column can

6、be taken out of the determinant.111212122212nnnnnnaaaaaakakaka nnnnnnaaaaaaaaa212222111211k=rik【 Corollary 2 】 If determinant D1 is obtained from the determinant D by multiplying every element in one row or in one column by a number k, then D1=kDMath. Dept., Wuhan University of Technology12252461235

7、10510121314314rr 2225552 5 Row (or column) i multiplies number k is marked as kri (or kci).Taking out the common cofactor k out of row i (or column i) is marked asor ,iirkckMath. Dept., Wuhan University of Technology【 Corollary 3 】 If a determinant has a row or column consisting entirely of zeros, t

8、he determinant is equal to zero. We can also get a new Corollary :246000314 2460 000314 0 Math. Dept., Wuhan University of Technology【 Corollary 4 】 if two rows or two columns of a determinant are proportional（成比例）（成比例）, then the determinant is equal to 0.1383902641183 330224 23c =0Math. Dept., Wuha

9、n University of Technology if each element of one row (column) say, the ith row, is the sum of two number, then the determinant is the sum of two determinants, such as:【Property 4】11121112212niiiiininnnnnaaabcbcbcaaa 111211112112121212nniiiniiinnnnnnnnnaaaaaabbbcccaaaaaa Math. Dept., Wuhan Universit

10、y of Technologyts.determinan of sum aas written becan t determinan then theelements, of sum is row one ofelement each ifSo,mm2465105314 2005105314 0405105314 0065105314 Math. Dept., Wuhan University of Technology【Property 5】 The determinant doesnt change when adding k times one row (or column) to an

11、other row (or column), marked as ri+krj or ci+kcj.nnnjninnjinjiaaaaaaaaaaaa12222111111nnnjninnjinjiaaaaaaaaaaaa12222111111nnnjninnjinjiaaaaaaaaaaaa12222111111kakakanjjj 21Add to column i12jjnjakakak Cj k Math. Dept., Wuhan University of TechnologyFor Example,3351110243152113 325111024215201331511002

12、41152113 =-1+0 1+2 0+1 1+2Math. Dept., Wuhan University of Technology121231213D 12122 132 212 1213 212rr 121121231242213213D=0=Math. Dept., Wuhan University of Technology11121314212223243132333441424344aaaaaaaaaaaaaaaa00000How to transform a determinant into an upper triangular determinant?0行列式计算的重要

13、方法：利用运算行列式计算的重要方法：利用运算ri+krj或或ci+kcj，把行列式，把行列式化为上（下）三角行列式化为上（下）三角行列式Math. Dept., Wuhan University of Technology 1121 2354 52749762 2512371459274612D 13ccD why?【Example 1】 Evaluate Solution:Math. Dept., Wuhan University of Technology 1522 1 73 429 5716 2421rr 312rr 41rr 0216 0113 0120 15220120 23rr 0

14、113 0216 第一列除第一行外的其余元素化为零第二行第二个元素使之变成1，因为最简单Math. Dept., Wuhan University of Technology15220113 322rr 42rr 0120 021 6 0030 0033152201130030 43rr 0033 00031 1 ( 3) 39 232rrMath. Dept., Wuhan University of Technology【Example 2 】Evaluate 234368abcdaabbccdDaabbccdaabbccd Solution:4332210023034rrabcdrrab

15、cDrrabcabc 4332000200abcdrrabcrrbcbc Math. Dept., Wuhan University of Technology430002000abcdabcrrbcc 2a bc From example 1 and example 2 we can see that: To evaluate a determinant, the most common method is to transform it into an upper triangular form, by using property 3,4,5.Math. Dept., Wuhan Uni

16、versity of Technology【Example 3 】Calculate nabbbbabbDbbabbbba The sum of ever row (or column) of the determinant is identical, so we can add column 2,3, to column 1,thenSolution:(1)(1)(1)(1)anbbbbanbabbanbbabanbbba 12ncccnD典型行列式，典型方法典型行列式，典型方法Math. Dept., Wuhan University of Technology 111(1)(1)11bb

17、babbcanbanbbabbba 213111000(1)000000nbbbabrranbabrrabrr 1(1)()nanb ab Math. Dept., Wuhan University of Technology 111112131nDn 【Example 4 】Evaluate “爪型行列式爪型行列式”111112131nDn Solution: 111111202030nn1213112131,nccccccn 21!(1)nini 典型行列式，典型方法典型行列式，典型方法Math. Dept., Wuhan University of TechnologySolution:

18、200134922D 232rr 20017109222 ( 1) 24 34222200Excercise 1 Calculate 134922D ：Math. Dept., Wuhan University of TechnologyActually, we have:1122 .DODD DD1360012300.111006785245673D 1213652123 ,73111DD12DD DProve it by yourself!Math. Dept., Wuhan University of TechnologyProperties you must remember: Int

19、erchanging two rows( or columns) of a determinant changes the sign of the determinant;Multiplying a single row or column of a determinant by a scalar has the effect of multiplying the value of the determinant by that scalar. Adding a multiple of one row (or column) to another does not change the value of the determinant.