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经济数学基础线性代数之第1章 行列式

Delegates, staff: Hello! in t he run-up to t he Spri ng Festival, we hel d one se ssion of four staff represe ntatives Conference 201 3-w orkshop, full ba ck in 20 12, careful analysi s of the current s ituation, di scuss 2 013 development pla ns. Here, on be half of my company 2 013 work re ports t o the Ge neral Assembl y, for consi deration. Pillar I, 2012 back i n 201 2, XX power compani es adhere to t he party's 1 7 great spirit for gui dance, compre hensively impleme nt the s cientific conce pt of devel opme nt, promoti ng cost-lea ders hip strategy, standards, focus on im plementation, lea n manageme nt, conti nuously im prove, sm ooth prese nt safety situation of enter prise management, busi ness ma nagement a nd control scie ntific and standardi zed, a nd the de dicati on of staff, manage a harmonious and dem ocratic atm osphere of the g ood situati on. Main i ndi cators are as foll ows :-t he battery indi cator: power ge neration t otaled 7. 815 billion kWh, bey ond the a nnual budget impleme ntation capa city of 315 milli on k Wh, a n increase of 757 milli on k Wh. Sales totaled 7.425 billi on kWh, exceeding sales of 330 million kWh the a nnual Executive budget, an i ncrea se of 729 milli on k Wh. --Se curity meas ures : unplanne d outage s 2.5 times. No personal inj ury accident occurred, no major a cci de nt and a bove , no maj or fire acci de nts without envir onme ntal pol lution accidents, safety for three conse cutive years to mai ntain sta bility to good post ure. – Business financial indicator s: total pr ofits of 255 milli on Yua n, beyond the a nnual budget of 207 milli on Y uan, beyond the Datang compa ny index 41.89 milli on Y uan, a n increase of 1.76 milli on Yua n, FCM a sses sment at grade four. --Energy: power s upply standard coal compl eting 312 .25 g /kWh, dow n 0.1 g /kWh; integrate d auxiliary power consum ption ratio in 5. 12%, dow n 0.26% ; polluta nt emissions performance greatly reduced compared to last yea r, car bon 0.09 g/k Wh, sulfur dioxide 0.104 g/k Wh NOx 0.5 12 g/k Wh; dust removal ef ficiency of more tha n 99.8 %. --Reliability i ndex: equivalent availa bility factor in 93.47% , increa sed 7.95% from a year earlie r. Equivalent force d outage rate 0. 08% , 0.16% reducti on ov er the same period a year earlier. Major achieveme nts: first, we s hould a dhere to the tw o "management system" basi s, strengt heni ng technological research, stre ngthen hidden hazar ds control a nd i ntrinsi c safety Enterpri se constr ucti on took new steps . -T he two "manag ement system" for improvement. Focus on promoting t he power of the compa ny manageme nt system and t he applicati on a nd impl ementation of the safety loop five-star management system, impr ove the s afety management system, realize the system of safety control. Furt her regulate s ecur ity routine s, safety supervisi on a nd ma nagement net work role t o play to a chieve closed -loop. Stre ngthe ning the supervisi on a nd manageme nt of ha bitual vi olation of, strengthe ning the safety supervisi on of outsourci ng contractor s. Carrie d out in s pring a nd a utumn of se curity i ns pecti ons , floo d contr ol and i ns pecti on, safety producti on month, day supervisi on of producti on safety and the Olympi c Games a nd ot her

第一单元
一、学习目标

行列式的定义

通过本节课学习,理解行列式的递归定义,掌握代数余子式的计算,知道任何 一个行列式就是代表一个数值,是可以经过特定的运算得到其结果的.

二、内容讲解
行列式 行列式的概念 什么叫做行列式呢?譬如,有 4 个数排列成一个行方块,在左右两边加竖线。

3 5 即 ? 1 2 称为二阶行列式;
有几个概念要清楚,即 上式中,横向称行,共有两行;竖向称列,共有两列; 一般用

aij

a12 ? 5 , a21 ? ?1 , 表示第 i 行第 j 列的元素, 如上例中的元素 a11 ? 3 ,

a22 ? 2 .
1 3 ?1 0 2 4

再看一个算式

5 ?7 0

称为三阶行列式,其中第三行为 5,-7,0;第二列为

–1,2,-7;元素 a23 ? 4 , a31 ? 5

0 2 3 1 1 0 3 0 4 ?1 ? 2 ? 3 1 0 ,是一个四阶行列式. 又如 0 0
而 a11 的代数余子式为
activities, compre hensive and tampi ng Safety Foundation ... Trouble shooti ng, manageme nt mechanism, give f ull play to r ole of techni cal s upervision and realizati on of troubleshooti ng, manag ement, impr oved pr oce ss management. T his year complete d the boiler low er hea der leakage, boiler pre ssure, a maj or risk ma nagement, compl eted 29 of great risks a nd 3 General problems of gover nance. Complete chemistry lab construction, thermal control, a nd complete t he boiler scal e integrated management, host s hafting vi bration of 10 s cientific and te chnol ogical proj ects, such a s. 势 趋 来 未 础 基 为 此 及 以 , 程 过 展 发 善 改 步 逐 到 生 产 从 、 了 述 简 先 首 ; 机 动 电 流 直 磁 永 是 的 绍 介 要 主 Complete supercriti cal 630 MW on -line simulati on system development a nd a ppli cation of cir culati ng water MCC standby power transformation, the t ransformation of de sulfuri zation waste water, the unit water s upply system of compre hensive treatment a nd discharge valve modification of coal mi ll 5 key technol ogical transformation proj ects, gr oup healt h is impr oved. --S cie nce a nd te chnol ogy innovation is furt her increase d. Strengt hen t he characteri stics of supercriti cal unit major is sue s,

gradually clearing t he particularity of s uper critical unit and regularit y. D eveloped m otor oil time manag ement, s witch acti on times , statistics, coal -aide d meas urement software, im proves t he pr oducti on l evel of lea n manag ement. I ncrea sed investment i n s cie nce and technolog y, reporting scie nce and technology proje cts a nd 14 techni cal proje ct total cost percentage of the total annual production output of 0.2 5%. "Large -s cale coal -fired power pla nt flue gas des ulfurizati on, de nitrification complete devel opment and applicati on of key technologie s" proje ct , w on t he national s cience a nd technol ogy progre ss second priz e. 63 0M W s uper critical units optimize d contr ol strategies and the 630M W development a nd application of on -line simul ation system for s uper critical unit s, s uper critical 600M W units of turbine driven boiler feed pum p set of comprehensiv e treatment of defects Datang technology re spe ctively one or two and thir d. Meanwhil e, informati on te chnol ogy achievement s, the company wa s named "China power informati on te chnol ogy be nchmarking enterprise s." --Repair and mainte nance ha s improved furt her. Modify the ins pe ction standards a nd sta ndar ds on a regular basis , standardizi ng work procedure s, checki ng and inspe ction proj ect. D eepe ning t he BFS++ system, and implements mai ntena nce i nformation share d. R eorgani zing RB l ogic again, and ensure the succe ss of the R B. Innovati ng the mechani sm of maintena nce ma nagement, impl emented a proje ct manager system. S uccessful completion of two aut onomous maintena nce, reliability improved steadil y. Impleme nting tw o c -level mainte nance, pr oject t hemselves 48.7% a nd 42.3%, res pective ly. A ccom plis h two cir culati ng pum ps repa ir and overha ul of four M ills, mainte nance teams to get exercis e. Promote the work of energy savi ng and cons umption reducing, com plete the unit e nergy cons umption diag nosis, pl ant water bala nce test, 10 e nergy-savi ng proj ects . Second, we s hould a dher e to "three" on t he economic benefits of improvi ng, outreach

Delegates, staff: Hello! in t he run-up to t he Spri ng Festival, we hel d one se ssion of four staff represe ntatives Conference 201 3-w orkshop, full ba ck in 20 12, careful analysi s of the current s ituation, di scuss 2 013 development pla ns. Here, on be half of my company 2 013 work re ports t o the Ge neral Assembl y, for consi deration. Pillar I, 2012 back i n 201 2, XX power compani es adhere to t he party's 1 7 great spirit for gui dance, compre hensively impleme nt the s cientific conce pt of devel opme nt, promoti ng cost-lea ders hip strategy, standards, focus on im plementation, lea n manageme nt, conti nuously im prove, sm ooth prese nt safety situation of enter prise management, busi ness ma nagement a nd control scie ntific and standardi zed, a nd the de dicati on of staff, manage a harmonious and dem ocratic atmosphere of the g ood situati on. Main i ndi cators are as foll ows :-t he battery indi cator: power ge neration t otaled 7. 815 billion kWh, bey ond the a nnual budget impleme ntation capa city of 315 milli on k Wh, a n increase of 757 milli on k Wh. Sales totaled 7.425 billi on kWh, exceeding sales of 330 million kWh the a nnual Executive budget, an i ncrea se of 729 milli on k Wh. --Se curity meas ures : unplanne d outage s 2.5 times. No personal inj ury accident occurred, no major a cci de nt and a bove , no maj or fire acci de nts without envir onme ntal pol lution accidents, safety for three conse cutive years to mai ntain sta bility to good post ure. – Business financial indicator s: total pr ofits of 255 milli on Yua n, beyond the a nnual budget of 207 milli on Y uan, beyond the Datang compa ny index 41.89 milli on Y uan, a n increase of 1.76 milli on Yua n, FCM a sses sment at grade four. --Energy: power s upply standard coal compl eting 312 .25 g /kWh, dow n 0.1 g /kWh; integrate d auxiliary power consum ption ratio in 5. 12%, dow n 0.26% ; polluta nt emissions performance greatly reduced compared to last yea r, car bon 0.09 g/k Wh, sulfur dioxide 0.104 g/k Wh NOx 0.5 12 g/k Wh; dust removal efficiency of more tha n 99.8 %. --Reliability i ndex: equivalent availa bility factor in 93.47% , increa sed 7.95% from a year earlie r. Equivalent force d outage rate 0. 08% , 0.16% reducti on ov er the same period a year earlier. Major achieveme nts: first, we s hould a dhere to the tw o "management system" basi s, strengt heni ng technological research, stre ngthen hidden hazar ds control a nd i ntrinsi c safety Enterpri se constr ucti on took new steps . -T he two "manag ement system" for improvement. Focus on promoting t he power of the compa ny manageme nt system and t he applicati on a nd impl ementation of the safety loop five-star management system, impr ove the s afety management system, realize the system of safety control. Furt her regulate s ecur ity routine s, safety supervisi on a nd ma nagement net work role t o play to a chieve closed -loop. Stre ngthe ning the supervisi on a nd manageme nt of ha bitual vi olation of, strengthe ning the safety supervisi on of outsourci ng contra ctor s. Carrie d out in s pring a nd a utumn of se curity i ns pecti ons , flood contr ol and i ns pecti on, safety producti on month, day supervisi on of producti on safety and the Olympi c Games a nd ot her

A11 ? ?? 1? M 11 ? ?
1?1

2 4 ?7 0

代数余子式就是在余子式前适当加正负号,正负号的规律是-1 的指数是该元 素的行数加列数.

A32 ? ?? 1?

3? 2

M 32 ? ?

1 0 3 4
Aij
是如何定义的?
i? j

问题思考: 元素

aij

的代数余子式

代数余子式

Aij

由符号因

i? j M a A ? ?? 1? 子 (?1) 与元素 ij 的余子式 ij 构成,即 ij

M ij

三、例题讲解
1 D? 3 ?2 ?4 0 4 2 ?3 5

例题 1:计算三阶行列式

分析:按照行列式的递归定义,将行列式的第一行展开,使它成为几个二阶行 列式之和, 二阶行列式可以利用对角相乘法,计算出结果.

D ? 1 ? ?? 1?
解:

1?1

0 ?3 ?3 0 1? 2 3 1? 3 3 ? ?? 4? ? ?? 1? ? 2 ? ?? 1? 4 5 ?2 5 ?2 4

? 1 ? 12 ? 4 ? 9 ? 2 ? 12 ? 72

四、课堂练习
a 0 0 0 c d D4 ? 0 e f g 0 0 计算行列式 b 0 0 h

利用 n 阶行列式的定义选择答案.

activities, compre hensive and tamping Safety Foundati on .. . Troubl eshooti ng, manageme nt mechanism, give full play to role of technical s upe rvision and realizati on of troubles hooting, ma nagement, impr oved process management. T his year completed t he boiler l ower hea der leakage, boil er pres sure, a major ri sk management, completed 29 of gr eat risks and 3 General problems of gover nance. Complete chemi stry lab constr ucti on, thermal control, a nd complete t he boiler s cale integrated ma nagement, host s hafting vi bration of 10 scie ntific and technologi cal pr ojects , such a s. 势 趋 来 未 础 基 为 此 及 以 , 程 过 展 发 善 改 步 逐 到 生 产 从 、 了 述 简 先 首 ; 机 动 电 流 直 磁 永 是 的 绍 介 要 主

Complete supercriti cal 630 MW on-line simulation system devel opme nt and applicati on of cir culati ng water MCC standby power transformation, t he transformation of de sulfuri zation w aste water, the unit water supply system of compre hensive treatment and dis charge valve modification of coal mill 5 key technologi cal transformation projects , group healt h is improve d. --S cience and technology i nnovati on is furt her increase d. Strengthen the characteristi cs of s uper critical unit major is sue s, Meanwhile, i nformation te chnol ogy achievements, t he company wa s named "China pow er information technol ogy be nchmarking

gradually cle aring the particularity of supercriti cal unit and regularity. Develope d motor oil time ma nagement, sw itch a ction times, statistics, coal-a ide d measureme nt software, impr oves the producti on lev el of lean management. I ncrea sed inve stment in s cie nce a nd technolog y, reporting s cie nce a nd technology proje cts and 14 techni cal pr oje ct total cost percentage of the total a nnual pr oduction output of 0.25% . "Large-s cale coal -fired power pla nt flue gas desulfur ization, de nitrification complete development and applicati on of key technologie s" proje ct , won the national sci ence and technology pr ogress s econd prize. 6 30M W supercriti cal units optimized contr ol strategies and the 6 30M W devel opme nt and application of on-li ne sim ulation system for supercritica l unit s, supercriti cal 600 MW units of turbine drive n bo iler feed pump set of compre hensive treatment of defect s Datang t echnol ogy respe ctively one or tw o and third.

enterprise s." --Repair and mainte nance ha s improved furt her. Modify the ins pe ction standards a nd sta ndar ds on a regular basis , standardizi ng work procedure s, checki ng and inspe ction proj ect. D eepe ning t he BFS++ system, and implements mai ntena nce i nformation share d. Reorgani zing RB l ogic again, and ensure the succe ss of the R B. Innovati ng the mechani sm of maintena nce ma nagement, impl emented a proje ct manager system. S uccessful completion of two aut onomous maintena nce, reliability improved steadil y. Impleme nting tw o c -level mainte nance, pr oject t hemselves 48.7% a nd 42.3%, res pective ly. A ccom plis h two cir culati ng pum ps repa ir and overha ul of four M ills, mainte nance teams to get exercis e.

Promote the work of energy savi ng and cons umption reducing, com plete the unit e nergy cons umptio n diag nosis, pl ant water bala nce test, 10 e nergy-savi ng proj ects .

Second, we s hould a dher e to "three" on t he economic benefits of improvi ng, outreach

2

Delegates, i nk the run-up t o the Spri ng Festival, wedheld one s essi on of four staff nce 2 013 -work shop, full back n 2012 , careful a nalysis the current sit uation, di scuss 20 13 dev elopment pla ns. Here, conse on be half of year my company 2 013 w orkity reports to posture the General Assembl y, for consi deration. Pillar I, 2012 ba inmillion 20 12, XX power compa nies a dher e to the 17 great pirit for guida comprehensively im plemmilli ent the entific conce pt of1.76 devel opment, promoti cost-lea ders hip strategy, focus on er imsupply plementati on, lea n manageme nt, continuously , smoot h pre sent safety situati power on of enterpri se manageme busine ss manag ement and scieperforma ntific andnce standardiz a nd the de dication of staff, a harmonious and democratic atm osphere of 0.512 the good Main i ndi cators are s follow s: -the battery indicator : power generati totale d 7.8 15 bill k Wh, t he annual budget implementati on capa city of forced 315 milli on k Wh, n increase of 757 million kWh. totaled billi on k Wh, exceedi ng sale s of 330 milli k Whathe annual Executive budget, ant n increase staff: of 729Hello! milli on Wh. --Security measures : unpla nne outages 2. 5 times. Norepresentatives persona l inj ury Confere accide nt occurred, no major accide nt iand above, no major fireof accidents wit hout environmental poll ution accide nts, s af ety for three cutive s to maintai n stabil to good . – Busines s financial i ndi cators: t otal profits of ck 255 Yua n, beyond the a nnual budget ofparty's 207 million Yuas n, beyond the nce, Datang company i ndex 4 1.89 on Ysci uan, an i ncreas e of mill ion Y uan, FCM ng as sessme nt at grade four. standards, --Energy: pow standard coal completi ng 312.25 g /k Wh,improve dow n 0.1 g/k Wh; integrated a uxiliary consumpti on rati o innt, 5. 12%, dow n 0.2 6%; polluta ntcontrol emissi ons greatly ed, reduced compare d to last year,manage car bon 0.09 g/k Wh, sulfur dioxide 0.10 4 g/k Wh NOx g/k situation. Wh; dust removal efficiency ofamore than 99 .8%. --Relia bi lity index: e quivale nton availability factor inion 93.47% , beyond increased 7.95% from a year e arlier. Equivalent outage ratea0.0 8%, 0. 16% reducti on over the s ame Sales peri od a year7.42 ear 5 lier. Major achieveme nts: first, we on s hould dhere to the two "manageme system" basis, stre ngthening technologi cal resear ch, stre ngthe n hi dde n hazar ds control and i ntrins ic safety Enter prise constructi on took ne w steps. -T he two "manageme nt system" for improvement. Focus on promoti ng the power of the com pany manageme nt system and the application a nd im pl ementation of the safety loop five-star manag ement system, improve t he safety management sy stem, realize the sy stem of safety control . Further reg ulate se curity routine s, safety supervisi on a nd manageme nt netw ork role to play to achieve cl ose d-l oop. Strengt heni ng the s upervision and manag ement of habit ual violati on of, strengtheni ng the s afety supervi sion of outsourcing contractors . Carrie d out i n spri ng and aut umn of se curity inspe ctions, flood contr ol and ins pe ction, safety production m onth, day supervisi on of production safety and the Olympi c Games a nd other

经济数学基础之线性代数

第 1 章 行列式

将行列式中的字母作为数字对待, 利用递归定义计算. 注意在该行列式的第一 行中,有两个零元素,因此展开式中对应的两项不用写出来了.
c e D4 = a ? (?1)1?1 ? 0 d f 0 0 0 c d f 0 0 ?0 e h b ? (?1)1?4 g 0 +

五、课后作业
1.求下列行列式的第二行第三列元素的代数余子式 A23
1 2 0 4 ?3 8 0 ?1 2

(1)

(2)

1 2 4 ?1 0 2 5 0

3 ?1 0 1 2 ?1 4 3

2.计算下列行列式
1 ?1 2 3 4 5 1

(1)

?2 6

1 0 1 (2) ? 2 2 1 4 0 1 0 5 0

0 ?1 0 2 4 ?2 3 5 0 1 0 6

1 2 D4 ? 3 1 3.设

0 0 1 0

(1)由定义计算 D4 ; (2)计算 a21 A21 ? a22 A22 ? a23 A23 ? a24 A24 ,即按第二行展开;

activities, compre hensive and tamping Safety Foundati on .. . Troubl eshooti ng, manageme nt mechanism, give full play to role of technical s upe rvision and realizati on of troubles hooting, ma nagement, impr oved process management. T his year completed t he boiler l ower hea der leakage, boi l er pres sure, a major ri sk management, completed 29 of great risks and 3 General problems of gover nance. Complete chemi stry lab constr ucti on, thermal control, a nd complete t he boiler s cale integrated ma nagement, host s hafting vi bration of 10 scie ntific and technologi cal pr ojects , such a s. 势 趋 来 未 础 基 为 此 及 以 , 程 过 展 发 善 改 步 逐 到 生 产 从 、 了 述 简 先 首 ; 机 动 电 流 直 磁 永 是 的 绍 介 要 主

Complete supercriti cal 630 MW on -line simulation system devel opme nt and applicati on of cir culati ng water MCC standby power transformation, t he transformation of de sulfuri zation w aste water, the unit water supply system of compre hensive treatment and dis charge valve modification of coal mill 5 key technologi cal transformation projects , group healt h is improve d. --S cience and technology i nnovati on is furt her increase d. Strengthen the characteristi cs of s uper critical unit major is sue s, Meanwhile, i nformation te chnol ogy achievements, t he company wa s named "China pow er information technol ogy be nchmarking

gradually cle aring the particularity of supercriti cal unit and regularity. Develope d motor oil time ma nagement, sw itch a ction times, statistics, coal-a ide d measureme nt software, impr oves the producti on lev el of lean management. I ncrea sed inve stment in s cie nce a nd technolog y, reporting s cie nce a nd technology proje cts and 14 techni cal pr oje ct total cost percentage of the tot al a nnual pr oduction output of 0.25% . "Large-s cale coal -fired power pla nt flue gas desulfur ization, de nitrification complete develo pment and applicati on of key technologie s" proje ct, won the national sci ence and technology pr ogress s econd prize. 6 30M W supercriti cal units optimized contr ol strategies and the 6 30M W devel opme nt and application of on-li ne sim ulation system for supercritica l unit s, supercriti cal 600 MW units of turbine drive n boiler feed pump set of compre hensive treatment of defect s Datang t echnol ogy respe ctively one or tw o and third.

enterprise s." --Repair and mainte nance ha s improved furt her. Modify the ins pe ction standards a nd sta ndar ds on a regular basis , standardizi ng work procedure s, checki ng and inspe ction proj ect. D eepe ning t he BFS++ system, and implements mai ntena nce i nformation share d. Reorgani zing RB l ogic again, and ensure the succe ss of the R B. Innovati ng the mecha ni sm of maintena nce ma nagement, impl emented a proje ct manager system. S uccessful completion of two aut onomous maintena nce, reliability improved steadil y. Impleme nting tw o c -level mainte nance, pr oject t hemselves 48.7% a nd 42.3%, res pective ly. A ccom plis h two cir culati ng pum ps repa ir and overha ul of four M ills, mainte nance teams to get exercis e.

Promote the work of energy savi ng and cons umption reducing, com plete the unit e nergy cons umption diag nosis, pl ant water bala nce test, 10 e nergy -savi ng proj ects .

Second, we s hould a dher e to "three" on t he economic benefits of improvi ng, outreach

——

3——

Delegates, staff: Hello! in t he run-up to t he Spri ng Festival, we hel d one se ssion of four staff represe ntatives Conference 201 3-w orkshop, full ba ck in 20 12, careful analysi s of the current s ituation, di scuss 2 013 development pla ns. Here, on be half of my company 2 013 work re ports t o the Ge neral Assembl y, for consi deration. Pillar I, 2012 back i n 201 2, XX power compani es adhere to t he party's 1 7 great spirit for gui dance, compre hensively impleme nt the s cientific conce pt of devel opme nt, promoti ng cost-lea ders hip strategy, standards, focus on im plementation, lea n manageme nt, conti nuously im prove, sm ooth prese nt safety situation of enter prise management, busi ness ma nagement a nd control scie ntific and standardi zed, a nd the de dicati on of staff, manage a harmonious and dem ocratic atmosphere of the g ood situati on. Main i ndi cators are as foll ows :-t he battery indi cator: power ge neration t otaled 7. 815 billion kWh, bey ond the a nnual budget impleme ntation capa city of 315 milli on k Wh, a n increase of 757 milli on k Wh. Sales totaled 7.425 billi on kWh, exceeding sales of 330 million kWh the a nnual Executive budget, an i ncrea se of 729 milli on k Wh. --Se curity meas ures : unplanne d outage s 2.5 times. No personal inj ury accident occurred, no major a cci de nt and a bove , no maj or fire acci de nts without envir onme ntal pol lution accidents, safety for three conse cutive years to mai ntain sta bility to good post ure. – Business financial indicator s: total pr ofits of 255 milli on Yua n, beyond the a nnual budget of 207 milli on Y uan, beyond the Datang compa ny index 41.89 milli on Y uan, a n increase of 1.76 milli on Yua n, FCM a sses sment at grade four. --Energy: power s upply standard coal compl eting 312 .25 g /kWh, dow n 0.1 g /kWh; integrate d auxiliary power consum ption ratio in 5. 12%, dow n 0.26% ; polluta nt emissions performance greatly reduced compared to last yea r, car bon 0.09 g/k Wh, sulfur dioxide 0.104 g/k Wh NOx 0.5 12 g/k Wh; dust removal efficiency of more tha n 99.8 %. --Reliability i ndex: equivalent availa bility factor in 93.47% , increa sed 7.95% from a year earlie r. Equivalent force d outage rate 0. 08% , 0.16% reducti on ov er the same period a year earlier. Major achieveme nts: first, we s hould a dhere to the tw o "management system" basi s, strengt heni ng technological research, stre ngthen hidden hazar ds control a nd i ntrinsi c safety Enterpri se constr ucti on took new steps . -T he two "manag ement system" for improvement. Focus on promoting t he power of the compa ny manageme nt system and t he applicati on a nd impl ementation of the safety loop five-star management system, impr ove the s afety management system, realize the system of safety control. Furt her regulate s ecur ity routine s, safety supervisi on a nd ma nagement net work role t o play to a chieve closed -loop. Stre ngthe ning the supervisi on a nd manageme nt of ha bitual vi olation of, strengthe ning the safety supervisi on of outsourci ng contra ctor s. Carrie d out in s pring a nd a utumn of se curity i ns pecti ons , flood contr ol and i ns pecti on, safety producti on month, day supervisi on of producti on safety and the Olympi c Games a nd ot her

(3)计算 a31 A31 ? a32 A32 ? a33 A33 ? a34 A34 ,即按第三行展开; (4)按第四行展开.
1 2 ?1

(?1) 2?3
1.(1) 2.(1)20 3.(1)1

1 2 0 ?1
(2)24 (2)1

( ?1)

2?3

(2)

0 2 ?1 5 0 3

(3)1

(4)1

第二单元
一、学习目标

行列式的性质

通过本节课的学习,掌握行列式的性质,并会利用这些性质计算行列式的值.

二、内容讲解 行列式的性质
用定义计算行列式的值有时是比较麻烦的,利用行列式的性质能够使计算变的 比较容易了. 行列式的性质有七条,下面讲一讲几条常用的性质.在讲这些性质前,先给出 一个概念: 把行列式 D 中的行与列按原顺序互换以后得到的行列式, 称为 D 的转置行列式, 记为 D .
1 2 3
T
T

1 4 7

D? 4 5 6 D ? 2 5 8 7 8 9 3 6 9 如 ,

3 4 3 5 ? ? ?2 5 6 4 6 1.行列式的行、列交换,其值不变.如
activities, compre hensive and tamping Safety Foundati on .. . Troubl eshooti ng, manageme nt mechanism, give full play to role of technical s upe rvision and realizati on of troubles hooting, ma nagement, impr oved process management. T his year completed t he boiler l ower hea der leakage, boil er pres sure, a major ri sk management, completed 29 of gr eat risks and 3 General problems of gover nance. Complete chemi stry lab constr ucti on, thermal control, a nd complete t he boiler s cale integrated ma nagement, host s hafting vi bration of 10 scie ntific and technologi cal pr ojects , such a s. 势 趋 来 未 础 基 为 此 及 以 , 程 过 展 发 善 改 步 逐 到 生 产 从 、 了 述 简 先 首 ; 机 动 电 流 直 磁 永 是 的 绍 介 要 主 Complete supercriti cal 630 MW on-line simulation system devel opme nt and applicati on of cir culati ng water MCC standby power transformation, t he transformation of de sulfuri zation w aste water, the unit water supply system of compre hensive treatment and dis charge valve modification of coal mill 5 key technologi cal transformation projects , group healt h is improve d. --S cience and technology i nnovati on is furt her increase d. Strengthen the characteristi cs of s uper critical unit major is sue s, Meanwhile, i nformation te chnol ogy achievements, t he company wa s named "China pow er information technol ogy be nchmarking gradually cle aring the particularity of supercriti cal unit and regularity. Develope d motor oil time ma nagement, sw itch a ction times, statistics, coal-a ide d measureme nt software, impr oves the producti on lev el of lean management. I ncrea sed inve stment in s cie nce a nd technolog y, reporting s cie nce a nd technology proje cts and 14 techni cal pr oje ct total cost percentage of the total a nnual pr oduction output of 0.25% . "Large-s cale coal -fired power pla nt flue gas desulfur ization, de nitrification complete development and applicati on of key technologie s" proje ct , won the national sci ence and technology pr ogress s econd prize. 6 30M W supercriti cal units optimized contr ol strategies and the 6 30M W devel opme nt and application of on-li ne sim ulation system for supercritica l unit s, supercriti cal 600 MW units of turbine drive n bo iler feed pump set of compre hensive treatment of defect s Datang t echnol ogy respe ctively one or tw o and third.

enterprise s." --Repair and mainte nance ha s improved furt her. Modify the ins pe ction standards a nd sta ndar ds on a regular basis , standardizi ng work procedure s, checki ng and inspe ction proj ect. D eepe ning t he BFS++ system, and implements mai ntena nce i nformation share d. Reorgani zing RB l ogic again, and ensure the succe ss of the R B. Innovati ng the mechani sm of maintena nce ma nagement, impl emented a proje ct manager system. S uccessful completion of two aut onomous maintena nce, reliability improved steadil y. Impleme nting tw o c -level mainte nance, pr oject t hemselves 48.7% a nd 42.3%, res pective ly. A ccom plis h two cir culati ng pum ps repa ir and overha ul of four M ills, mainte nance teams to get exercis e.

Promote the work of energy savi ng and cons umption reducing, com plete the unit e nergy cons umptio n diag nosis, pl ant water bala nce test, 10 e nergy-savi ng proj ects .

Second, we s hould a dher e to "three" on t he economic benefits of improvi ng, outreach

4

Delegates, i nk the run-up t o the Spri ng Festival, wedheld one s essi on of four staff nce 2 013 -work shop, full back n 2012 , careful a nalysis the current sit uation, di scuss 20 13 dev elopment pla ns. Here, conse on be half of year my company 2 013 w orkity reports to posture the General Assembl y, for consi deration. Pillar I, 2012 ba inmillion 20 12, XX power compa nies a dher e to the 17 great pirit for guida comprehensively im plemmilli ent the entific conce pt of1.76 devel opment, promoti cost-lea ders hip strategy, focus on er imsupply plementati on, lea n manageme nt, continuously , smoot h pre sent safety situati power on of enterpri se manageme busine ss manag ement and scieperforma ntific andnce standardiz a nd the de dication of staff, a harmonious and democratic atm osphere of 0.512 the good Main i ndi cators are s follow s: -the battery indicator : power generati totale d 7.8 15 bill k Wh, t he annual budget implementati on capa city of forced 315 milli on k Wh, n increase of 757 million kWh. totaled billi on k Wh, exceedi ng sale s of 330 milli k Whathe annual Executive budget, ant n increase staff: of 729Hello! milli on Wh. --Security measures : unpla nne outages 2. 5 times. Norepresentatives persona l inj ury Confere accide nt occurred, no major accide nt iand above, no major fireof accidents wit hout environmental poll ution accide nts, s af ety for three cutive s to maintai n stabil to good . – Busines s financial i ndi cators: t otal profits of ck 255 Yua n, beyond the a nnual budget ofparty's 207 million Yuas n, beyond the nce, Datang company i ndex 4 1.89 on Ysci uan, an i ncreas e of mill ion Y uan, FCM ng as sessme nt at grade four. standards, --Energy: pow standard coal completi ng 312.25 g /k Wh,improve dow n 0.1 g/k Wh; integrated a uxiliary consumpti on rati o innt, 5. 12%, dow n 0.2 6%; polluta ntcontrol emissi ons greatly ed, reduced compare d to last year,manage car bon 0.09 g/k Wh, sulfur dioxide 0.10 4 g/k Wh NOx g/k situation. Wh; dust removal efficiency ofamore than 99 .8%. --Relia bi lity index: e quivale nton availability factor inion 93.47% , beyond increased 7.95% from a year e arlier. Equivalent outage ratea0.0 8%, 0. 16% reducti on over the s ame Sales peri od a year7.42 ear 5 lier. Major achieveme nts: first, we on s hould dhere to the two "manageme system" basis, stre ngthening technologi cal resear ch, stre ngthe n hi dde n hazar ds control and i ntrins ic safety Enter prise constructi on took ne w steps. -T he two "manageme nt system" for improvement. Focus on promoti ng the power of the com pany manageme nt system and the application a nd im pl ementation of the safety loop five-star manag ement system, improve t he safety management sy stem, realize the sy stem of safety control . Further reg ulate se curity routine s, safety supervisi on a nd manageme nt netw ork role to play to achieve cl ose d-l oop. Strengt heni ng the s upervision and manag ement of habit ual violati on of, strengtheni ng the s afety supervi sion of outsourcing contractors . Carrie d out i n spri ng and aut umn of se curity inspe ctions, flood contr ol and ins pe ction, safety production m onth, day supervisi on of production safety and the Olympi c Games a nd other

经济数学基础之线性代数
这条性质说明行列式中,行与列的地位是一样的.

第 1 章 行列式

3 4 5 6 ?? ? ?2 5 6 3 4 2.行列式的两行交换,其值变号.如 a b a b ?3 c d 3.若行列式的某一行有公因子,则可提出.如 3c 3d
注意:一个行列式与一个数相乘,等于该数与行列式的某行(列)的元素相乘. 4.行列式对行的倍加运算,其值不变.如倍加运算就是把一行的常数倍加到

3 1 ?+2? 另一行上 ? 1 ? 2

3 1 ? ?5 5 0

注意:符号“?+2?”放在等号上面,表示行变换,放在等号下面表示列变换. 问题 1:将 n 阶行列式的最后一行轮换到第一行, 这两个行列式的值有什么关 系?答案设 n 阶行列式 Dn ,若将 Dn 的最后一行轮换到第一行,得另一个 n 阶行列
n?1 式 C n ,那么这两个行列式的值的关系为: C n = (?1) Dn

问题 2:如果行列式有两行或两行以上的行都有公因子,那么按性质 3 应如何 提取? 答案按顺序将公因子提出. 三、例题讲解

a 0 4 b ?1 8 例 1 计算行列式 5 ? 7

0 0 c 6

0 0 0 d .

分析:利用性质 6,行列式可以按任一行(列)展开.本题按第一行逐步展开, 计算出结果.

activities, compre hensive and tamping Safety Foundati on .. . Troubl eshooti ng, manageme nt mechanism, give full play to role of technical s upe rvision and realizati on of troubles hooting, ma nagement, impr oved process management. T his year completed t he boiler l ower hea der leakage, boi l er pres sure, a major ri sk management, completed 29 of great risks and 3 General problems of gover nance. Complete chemi stry lab constr ucti on, thermal control, a nd complete t he boiler s cale integrated ma nagement, host s hafting vi bration of 10 scie ntific and technologi cal pr ojects , such a s. 势 趋 来 未 础 基 为 此 及 以 , 程 过 展 发 善 改 步 逐 到 生 产 从 、 了 述 简 先 首 ; 机 动 电 流 直 磁 永 是 的 绍 介 要 主

Complete supercriti cal 630 MW on -line simulation system devel opme nt and applicati on of cir culati ng water MCC standby power transformation, t he transformation of de sulfuri zation w aste water, the unit water supply system of compre hensive treatment and dis charge valve modification of coal mill 5 key technologi cal transformation projects , group healt h is improve d. --S cience and technology i nnovati on is furt her increase d. Strengthen the characteristi cs of s uper critical unit major is sue s, Meanwhile, i nformation te chnol ogy achievements, t he company wa s named "China pow er information technol ogy be nchmarking

gradually cle aring the particularity of supercriti cal unit and regularity. Develope d motor oil time ma nagement, sw itch a ction times, statistics, coal-a ide d measureme nt software, impr oves the producti on lev el of lean management. I ncrea sed inve stment in s cie nce a nd technolog y, reporting s cie nce a nd technology proje cts and 14 techni cal pr oje ct total cost percentage of the tot al a nnual pr oduction output of 0.25% . "Large-s cale coal -fired power pla nt flue gas desulfur ization, de nitrification complete develo pment and applicati on of key technologie s" proje ct, won the national sci ence and technology pr ogress s econd prize. 6 30M W supercriti cal units optimized contr ol strategies and the 6 30M W devel opme nt and application of on-li ne sim ulation system for supercritica l unit s, supercriti cal 600 MW units of turbine drive n boiler feed pump set of compre hensive treatment of defect s Datang t echnol ogy respe ctively one or tw o and third.

enterprise s." --Repair and mainte nance ha s improved furt her. Modify the ins pe ction standards a nd sta ndar ds on a regular basis , standardizi ng work procedure s, checki ng and inspe ction proj ect. D eepe ning t he BFS++ system, and implements mai ntena nce i nformation share d. Reorgani zing RB l ogic again, and ensure the succe ss of the R B. Innovati ng the mecha ni sm of maintena nce ma nagement, impl emented a proje ct manager system. S uccessful completion of two aut onomous maintena nce, reliability improved steadil y. Impleme nting tw o c -level mainte nance, pr oject t hemselves 48.7% a nd 42.3%, res pective ly. A ccom plis h two cir culati ng pum ps repa ir and overha ul of four M ills, mainte nance teams to get exercis e.

Promote the work of energy savi ng and cons umption reducing, com plete the unit e nergy cons umption diag nosis, pl ant water bala nce test, 10 e nergy -savi ng proj ects .

Second, we s hould a dher e to "three" on t he economic benefits of improvi ng, outreach

——

5——

Delegates, staff: Hello! in t he run-up to t he Spri ng Festival, we hel d one se ssion of four staff represe ntatives Conference 201 3-w orkshop, full ba ck in 20 12, careful analysi s of the current s ituation, di scuss 2 013 development pla ns. Here, on be half of my company 2 013 work re ports t o the Ge neral Assembl y, for consi deration. Pillar I, 2012 back i n 201 2, XX power compani es adhere to t he party's 1 7 great spirit for gui dance, compre hensively impleme nt the s cientific conce pt of devel opme nt, promoti ng cost-lea ders hip strategy, standards, focus on im plementation, lea n manageme nt, conti nuously im prove, sm ooth prese nt safety situation of enter prise management, busi ness ma nagement a nd control scie ntific and standardi zed, a nd the de dicati on of staff, manage a harmonious and dem ocratic atmosphere of the g ood situati on. Main i ndi cators are as foll ows :-t he battery indi cator: power ge neration t otaled 7. 815 billion kWh, bey ond the a nnual budget impleme ntation capa city of 315 milli on k Wh, a n increase of 757 milli on k Wh. Sales totaled 7.425 billi on kWh, exceeding sales of 330 million kWh the a nnual Executive budget, an i ncrea se of 729 milli on k Wh. --Se curity meas ures : unplanne d outage s 2.5 times. No personal inj ury accident occurred, no major a cci de nt and a bove , no maj or fire acci de nts without envir onme ntal pol lution accidents, safety for three conse cutive years to mai ntain sta bility to good post ure. – Business financial indicator s: total pr ofits of 255 milli on Yua n, beyond the a nnual budget of 207 milli on Y uan, beyond the Datang compa ny index 41.89 milli on Y uan, a n increase of 1.76 milli on Yua n, FCM a sses sment at grade four. --Energy: power s upply standard coal compl eting 312 .25 g /kWh, dow n 0.1 g /kWh; integrate d auxiliary power consum ption ratio in 5. 12%, dow n 0.26% ; polluta nt emissions performance greatly reduced compared to last yea r, car bon 0.09 g/k Wh, sulfur dioxide 0.104 g/k Wh NOx 0.5 12 g/k Wh; dust removal efficiency of more tha n 99.8 %. --Reliability i ndex: equivalent availa bility factor in 93.47% , increa sed 7.95% from a year earlie r. Equivalent force d outage rate 0. 08% , 0.16% reducti on ov er the same period a year earlier. Major achieveme nts: first, we s hould a dhere to the tw o "management system" basi s, strengt heni ng technological research, stre ngthen hidden hazar ds control a nd i ntrinsi c safety Enterpri se constr ucti on took new steps . -T he two "manag ement system" for improvement. Focus on promoting t he power of the compa ny manageme nt system and t he applicati on a nd impl ementation of the safety loop five-star management system, impr ove the s afety management system, realize the system of safety control. Furt her regulate s ecur ity routine s, safety supervisi on a nd ma nagement net work role t o play to a chieve closed -loop. Stre ngthe ning the supervisi on a nd manageme nt of ha bitual vi olation of, strengthe ning the safety supervisi on of outsourci ng contra ctor s. Carrie d out in s pring a nd a utumn of se curity i ns pecti ons , flood contr ol and i ns pecti on, safety producti on month, day supervisi on of producti on safety and the Olympi c Games a nd ot her

a 0 4 b ?1 8 解: 5 ? 7

0 0 c 6

0 b 0 0 0 c 0 0 a 8 c 0 ab d = ? 7 6 d = 6 d = abcd

我们将行列式中由左上角至右下角的对角线, 称为主对角线.如例 1 中,行列 式在主对角线以上的元素全为零,则称为下三角行列式. 由例 1 的计算过程,可得 这样规律:下三角行列式就等于主对角线元素的积. 同理,主对角线以下元素全为 零的行列式,则称为上三角行列式,且上三角行列式也等于主对角线元素之积.今 后,上、下三角行列式统称为三角行列式.

1 8 2 例 2 计算行列式 ? 7

2 3 4 9 ?7 ?6 4 6 8 7 9 ?4

分析:原行列式中第三行的元素是第一行的 2 倍,因此,利用行列式的倍加运 算(性质 5),使第三行的元素都变为 0,得到行列式的值.

1 8 2 解: ? 7

2 3 4 9 ?7 ?6 4 6 8 7 9 ?4

1 8 0 ?7

2 3 4 9 ?7 ?6 0 0 0 7 9 ?4 = 0

1 ?1 2 ? 2 4 ? 3 ?1 1 0 2 3 1 2 2 例 3 计算行列式 1 1
分析:利用行列式的倍加运算(性质 5),首先将某行(列)的元素尽可能化 为 0,再利用行列式可以按任一行(列)展开的性质(性质 6),逐步将原行列式化 为二阶行列式,计算出结果.

activities, compre hensive and tamping Safety Foundati on .. . Troubl eshooti ng, manageme nt mechanism, give full play to role of technical s upe rvision and realizati on of troubles hooting, ma nagement, impr oved process management. T his year completed t he boiler l ower hea der leakage, boil er pres sure, a major ri sk management, completed 29 of gr eat risks and 3 General problems of gover nance. Complete chemi stry lab constr ucti on, thermal control, a nd complete t he boiler s cale integrated ma nagement, host s hafting vi bration of 10 scie ntific and technologi cal pr ojects , such a s. 势 趋 来 未 础 基 为 此 及 以 , 程 过 展 发 善 改 步 逐 到 生 产 从 、 了 述 简 先 首 ; 机 动 电 流 直 磁 永 是 的 绍 介 要 主

Complete supercriti cal 630 MW on-line simulation system devel opme nt and applicati on of cir culati ng water MCC standby power transformation, t he transformation of de sulfuri zation w aste water, the unit water supply system of compre hensive treatment and dis charge valve modification of coal mill 5 key technologi cal transformation projects , group healt h is improve d. --S cience and technology i nnovati on is furt her increase d. Strengthen the characteristi cs of s uper critical unit major is sue s, Meanwhile, i nformation te chnol ogy achievements, t he company wa s named "China pow er information technol ogy be nchmarking

gradually cle aring the particularity of supercriti cal unit and regularity. Develope d motor oil time ma nagement, sw itch a ction times, statistics, coal-a ide d measureme nt software, impr oves the producti on lev el of lean management. I ncrea sed inve stment in s cie nce a nd technolog y, reporting s cie nce a nd technology proje cts and 14 techni cal pr oje ct total cost percentage of the total a nnual pr oduction output of 0.25% . "Large-s cale coal -fired power pla nt flue gas desulfur ization, de nitrification complete development and applicati on of key technologie s" proje ct , won the national sci ence and technology pr ogress s econd prize. 6 30M W supercriti cal units optimized contr ol strategies and the 6 30M W devel opme nt and application of on-li ne sim ulation system for supercritica l unit s, supercriti cal 600 MW units of turbine drive n bo iler feed pump set of compre hensive treatment of defect s Datang t echnol ogy respe ctively one or tw o and third.

enterprise s." --Repair and mainte nance ha s improved furt her. Modify the ins pe ction standards a nd sta ndar ds on a regular basis , standardizi ng work procedure s, checki ng and inspe ction proj ect. D eepe ning t he BFS++ system, and implements mai ntena nce i nformation share d. Reorgani zing RB l ogic again, and ensure the succe ss of the R B. Innovati ng the mechani sm of maintena nce ma nagement, impl emented a proje ct manager system. S uccessful completion of two aut onomous maintena nce, reliability improved steadil y. Impleme nting tw o c -level mainte nance, pr oject t hemselves 48.7% a nd 42.3%, res pective ly. A ccom plis h two cir culati ng pum ps repa ir and overha ul of four M ills, mainte nance teams to get exercis e.

Promote the work of energy savi ng and cons umption reducing, com plete the unit e nergy cons umptio n diag nosis, pl ant water bala nce test, 10 e nergy-savi ng proj ects .

Second, we s hould a dher e to "three" on t he economic benefits of improvi ng, outreach

6

Delegates, i nk the run-up t o the Spri ng Festival, wedheld one s essi on of four staff nce 2 013 -work shop, full back n 2012 , careful a nalysis the current sit uation, di scuss 20 13 dev elopment pla ns. Here, conse on be half of year my company 2 013 w orkity reports to posture the General Assembl y, for consi deration. Pillar I, 2012 ba inmillion 20 12, XX power compa nies a dher e to the 17 great pirit for guida comprehensively im plemmilli ent the entific conce pt of1.76 devel opment, promoti cost-lea ders hip strategy, focus on er imsupply plementati on, lea n manageme nt, continuously , smoot h pre sent safety situati power on of enterpri se manageme busine ss manag ement and scieperforma ntific andnce standardiz a nd the de dication of staff, a harmonious and democratic atm osphere of 0.512 the good Main i ndi cators are s follow s: -the battery indicator : power generati totale d 7.8 15 bill k Wh, t he annual budget implementati on capa city of forced 315 milli on k Wh, n increase of 757 million kWh. totaled billi on k Wh, exceedi ng sale s of 330 milli k Whathe annual Executive budget, ant n increase staff: of 729Hello! milli on Wh. --Security measures : unpla nne outages 2. 5 times. Norepresentatives persona l inj ury Confere accide nt occurred, no major accide nt iand above, no major fireof accidents wit hout environmental poll ution accide nts, s af ety for three cutive s to maintai n stabil to good . – Busines s financial i ndi cators: t otal profits of ck 255 Yua n, beyond the a nnual budget ofparty's 207 million Yuas n, beyond the nce, Datang company i ndex 4 1.89 on Ysci uan, an i ncreas e of mill ion Y uan, FCM ng as sessme nt at grade four. standards, --Energy: pow standard coal completi ng 312.25 g /k Wh,improve dow n 0.1 g/k Wh; integrated a uxiliary consumpti on rati o innt, 5. 12%, dow n 0.2 6%; polluta ntcontrol emissi ons greatly ed, reduced compare d to last year,manage car bon 0.09 g/k Wh, sulfur dioxide 0.10 4 g/k Wh NOx g/k situation. Wh; dust removal efficiency ofamore than 99 .8%. --Relia bi lity index: e quivale nton availability factor inion 93.47% , beyond increased 7.95% from a year e arlier. Equivalent outage ratea0.0 8%, 0. 16% reducti on over the s ame Sales peri od a year7.42 ear 5 lier. Major achieveme nts: first, we on s hould dhere to the two "manageme system" basis, stre ngthening technologi cal resear ch, stre ngthe n hi dde n hazar ds control and i ntrins ic safety Enter prise constructi on took ne w steps. -T he two "manageme nt system" for improvement. Focus on promoti ng the power of the com pany manageme nt system and the application a nd im pl ementation of the safety loop five-star manag ement system, improve t he safety management sy stem, realize the sy stem of safety control . Further reg ulate se curity routine s, safety supervisi on a nd manageme nt netw ork role to play to achieve cl ose d-l oop. Strengt heni ng the s upervision and manag ement of habit ual violati on of, strengtheni ng the s afety supervi sion of outsourcing contractors . Carrie d out i n spri ng and aut umn of se curity inspe ctions, flood contr ol and ins pe ction, safety production m onth, day supervisi on of production safety and the Olympi c Games a nd other

经济数学基础之线性代数
1 ?1 2 ? 2 4 ? 3 ?1 1 0 2 3 1 2 2 解: 1 1
?+?

第 1 章 行列式
1 ?1 0 ? 2 4 ?3 0 1 0 2 4 1 1 1 4 2

1 ?1 0 ? 2 1 ?1 ? 2 4 ?3 0 1 3? 3 0 2 4 1 4 ? ( ?1) 4 ? 3 1 1 ?1 1 1 ?1 0 1 =
0 ?1 ? 2
?+?

4? 1 ? 3 0 ?1

1 1

4 ? (?1) 2?1
=

?1 ? 2 ? 1 1 ? ?4(?1 ? 2) ? 12

通过此例可知,行列式两行成比例,则行列式为零.

三、课堂练习
a11 a 21 a12 a 22 a 32 a13 a 23 ? d a 33 3a 31 ? a11 3a 32 ? a12 2a 22 3a 31 ? a13 2a 23

练习 1 若

a 31

,求行列式

2a 21

利用行列式的性质 3,将第一行的公因子 3、第二行的公因子(-1)、第三行的 公因子 2 提出. 利用行列式的性质 3 和性质 2,将所要计算的行列式化为已知的行列式,再求 其值.

1 9 2 ?8 3 7 练习 2 计算行列式 4 ? 5

103 ? 3 198 2 299 1 405 ? 5

由性质 4,若行列式中某列的元素均为两项之和,则可将其拆写成两个行列式 之和.
activities, compre hensive and tamping Safety Foundati on .. . Troubl eshooti ng, manageme nt mechanism, give full play to role of technical s upe rvision and realizati on of troubles hooting, ma nagement, impr oved process management. T his year completed t he boiler l ower hea der leakage, boi l er pres sure, a major ri sk management, completed 29 of great risks and 3 General problems of gover nance. Complete chemi stry lab constr ucti on, thermal control, a nd complete t he boiler s cale integrated ma nagement, host s hafting vi bration of 10 scie ntific and technologi cal pr ojects , such a s. 势 趋 来 未 础 基 为 此 及 以 , 程 过 展 发 善 改 步 逐 到 生 产 从 、 了 述 简 先 首 ; 机 动 电 流 直 磁 永 是 的 绍 介 要 主 Complete supercriti cal 630 MW on -line simulation system devel opme nt and applicati on of cir culati ng water MCC standby power transformation, t he transformation of de sulfuri zation w aste water, the unit water supply system of compre hensive treatment and dis charge valve modification of coal mill 5 key technologi cal transformation projects , group healt h is improve d. --S cience and technology i nnovati on is furt her increase d. Strengthen the characteristi cs of s uper critical unit major is sue s, Meanwhile, i nformation te chnol ogy achievements, t he company wa s named "China pow er information technol ogy be nchmarking gradually cle aring the particularity of supercriti cal unit and regularity. Develope d motor oil time ma nagement, sw itch a ction times, statistics, coal-a ide d measureme nt software, impr oves the producti on lev el of lean management. I ncrea sed inve stment in s cie nce a nd technolog y, reporting s cie nce a nd technology proje cts and 14 techni cal pr oje ct total cost percentage of the tot al a nnual pr oduction output of 0.25% . "Large-s cale coal -fired power pla nt flue gas desulfur ization, de nitrification complete develo pment and applicati on of key technologie s" proje ct, won the national sci ence and technology pr ogress s econd prize. 6 30M W supercriti cal units optimized contr ol strategies and the 6 30M W devel opme nt and application of on-li ne sim ulation system for supercritica l unit s, supercriti cal 600 MW units of turbine drive n boiler feed pump set of compre hensive treatment of defect s Datang t echnol ogy respe ctively one or tw o and third. enterprise s." --Repair and mainte nance ha s improved furt her. Modify the ins pe ction standards a nd sta ndar ds on a regular basis , standardizi ng work procedure s, checki ng and inspe ction proj ect. D eepe ning t he BFS++ system, and implements mai ntena nce i nformation share d. Reorgani zing RB l ogic again, and ensure the succe ss of the R B. Innovati ng the mecha ni sm of maintena nce ma nagement, impl emented a proje ct manager system. S uccessful completion of two aut onomous maintena nce, reliability improved steadil y. Impleme nting tw o c -level mainte nance, pr oject t hemselves 48.7% a nd 42.3%, res pective ly. A ccom plis h two cir culati ng pum ps repa ir and overha ul of four M ills, mainte nance teams to get exercis e. Promote the work of energy savi ng and cons umption reducing, com plete the unit e nergy cons umption diag nosis, pl ant water bala nce test, 10 e nergy -savi ng proj ects . Second, we s hould a dher e to "three" on t he economic benefits of improvi ng, outreach

——

7——

Delegates, staff: Hello! in t he run-up to t he Spri ng Festival, we hel d one se ssion of four staff represe ntatives Conference 201 3-w orkshop, full ba ck in 20 12, careful analysi s of the current s ituation, di scuss 2 013 development pla ns. Here, on be half of my company 2 013 work re ports t o the Ge neral Assembl y, for consi deration. Pillar I, 2012 back i n 201 2, XX power compani es adhere to t he party's 1 7 great spirit for gui dance, compre hensively impleme nt the s cientific conce pt of devel opme nt, promoti ng cost-lea ders hip strategy, standards, focus on im plementation, lea n manageme nt, conti nuously im prove, sm ooth prese nt safety situation of enter prise management, busi ness ma nagement a nd control scie ntific and standardi zed, a nd the de dicati on of staff, manage a harmonious and dem ocratic atmosphere of the g ood situati on. Main i ndi cators are as foll ows :-t he battery indi cator: power ge neration t otaled 7. 815 billion kWh, bey ond the a nnual budget impleme ntation capa city of 315 milli on k Wh, a n increase of 757 milli on k Wh. Sales totaled 7.425 billi on kWh, exceeding sales of 330 million kWh the a nnual Executive budget, an i ncrea se of 729 milli on k Wh. --Se curity meas ures : unplanne d outage s 2.5 times. No personal inj ury accident occurred, no major a cci de nt and a bove , no maj or fire acci de nts without envir onme ntal pol lution accidents, safety for three conse cutive years to mai ntain sta bility to good post ure. – Business financial indicator s: total pr ofits of 255 milli on Yua n, beyond the a nnual budget of 207 milli on Y uan, beyond the Datang compa ny index 41.89 milli on Y uan, a n increase of 1.76 milli on Yua n, FCM a sses sment at grade four. --Energy: power s upply standard coal compl eting 312 .25 g /kWh, dow n 0.1 g /kWh; integrate d auxiliary power consum ption ratio in 5. 12%, dow n 0.26% ; polluta nt emissions performance greatly reduced compared to last yea r, car bon 0.09 g/k Wh, sulfur dioxide 0.104 g/k Wh NOx 0.5 12 g/k Wh; dust removal efficiency of more tha n 99.8 %. --Reliability i ndex: equivalent availa bility factor in 93.47% , increa sed 7.95% from a year earlie r. Equivalent force d outage rate 0. 08% , 0.16% reducti on ov er the same period a year earlier. Major achieveme nts: first, we s hould a dhere to the tw o "management system" basi s, strengt heni ng technological research, stre ngthen hidden hazar ds control a nd i ntrinsi c safety Enterpri se constr ucti on took new steps . -T he two "manag ement system" for improvement. Focus on promoting t he power of the compa ny manageme nt system and t he applicati on a nd impl ementation of the safety loop five-star management system, impr ove the s afety management system, realize the system of safety control. Furt her regulate s ecur ity routine s, safety supervisi on a nd ma nagement net work role t o play to a chieve closed -loop. Stre ngthe ning the supervisi on a nd manageme nt of ha bitual vi olation of, strengthe ning the safety supervisi on of outsourci ng contra ctor s. Carrie d out in s pring a nd a utumn of se curity i ns pecti ons , flood contr ol and i ns pecti on, safety producti on month, day supervisi on of producti on safety and the Olympi c Games a nd ot her

在着手具体计算前, 先观察一下此行列式有否特点?有, 其第三列的数字较大, 但又都分别接近 100、200、300 和 400,故将第三列的元素分别写成两项之和, 再 利用行列式的性质 4 将其写成两个行列式之和.注意,将第三列的元素分别写成两 项之和时,还要考虑到结论“行列式中两列元素相同(或成比例),则该行列式的 值为 0”的利用.

五、课后作业
1.计算下列行列式
0 ?1 ? 5 7 0 1 2 ?1 2 3 3 5 1 2 1 0 5 ?7

(1)

(2)
1

1

1

(3) 2.证明

w w w w2

w2 w

(w ? 0 )
c?a

1 2 2 4 0 0 (4) 9 ? 7
a2 1

3 4 7 8 1 0 8 ?3

a ?b b?c

(1)

b?c c?a a?b ? 0 c?a a ?b b?c

2a a ? b 2b ? ?a ? b ?

ab 1

b2 1

3

(2)

1. (1)0

(2) -2

2 2 (3) ? w (w ? 1)

(4)0

2. (1)提示:利用性质 5,将第一行化成零行. (2)提示:利用性质 5,将第三行的元素化成“0 0 1” ,再按第三行展开, 并推出等号右边结果.

第三单元
一、学习目标

行列式的计算

activities, compre hensive and tamping Safety Foundati on .. . Troubl eshooti ng, manageme nt mechanism, give full play to role of technical s upe rvision and realizati on of troubles hooting, ma nagement, impr oved process management. T his year completed t he boiler l ower hea der leakage, boil er pres sure, a major ri sk management, completed 29 of gr eat risks and 3 General problems of gover nance. Complete chemi stry lab constr ucti on, thermal control, a nd complete t he boiler s cale integrated ma nagement, host s hafting vi bration of 10 scie ntific and technologi cal pr ojects , such a s. 势 趋 来 未 础 基 为 此 及 以 , 程 过 展 发 善 改 步 逐 到 生 产 从 、 了 述 简 先 首 ; 机 动 电 流 直 磁 永 是 的 绍 介 要 主

Complete supercriti cal 630 MW on-line simulation system devel opme nt and applicati on of cir culati ng water MCC standby power transformation, t he transformation of de sulfuri zation w aste water, the unit water supply system of compre hensive treatment and dis charge valve modification of coal mill 5 key technologi cal transformation projects , group healt h is improve d. --S cience and technology i nnovati on is furt her increase d. Strengthen the characteristi cs of s uper critical unit major is sue s, Meanwhile, i nformation te chnol ogy achievements, t he company wa s named "China pow er information technol ogy be nchmarking

gradually cle aring the particularity of supercriti cal unit and regularity. Develope d motor oil time ma nagement, sw itch a ction times, statistics, coal-a ide d measureme nt software, impr oves the producti on lev el of lean management. I ncrea sed inve stment in s cie nce a nd technolog y, reporting s cie nce a nd technology proje cts and 14 techni cal pr oje ct total cost percentage of the total a nnual pr oduction output of 0.25% . "Large-s cale coal -fired power pla nt flue gas desulfur ization, de nitrification complete development and applicati on of key technologie s" proje ct , won the national sci ence and technology pr ogress s econd prize. 6 30M W supercriti cal units optimized contr ol strategies and the 6 30M W devel opme nt and application of on-li ne sim ulation system for supercritica l unit s, supercriti cal 600 MW units of turbine drive n bo iler feed pump set of compre hensive treatment of defect s Datang t echnol ogy respe ctively one or tw o and third.

enterprise s." --Repair and mainte nance ha s improved furt her. Modify the ins pe ction standards a nd sta ndar ds on a regular basis , standardizi ng work procedure s, checki ng and inspe ction proj ect. D eepe ning t he BFS++ system, and implements mai ntena nce i nformation share d. Reorgani zing RB l ogic again, and ensure the succe ss of the R B. Innovati ng the mechani sm of maintena nce ma nagement, impl emented a proje ct manager system. S uccessful completion of two aut onomous maintena nce, reliability improved steadil y. Impleme nting tw o c -level mainte nance, pr oject t hemselves 48.7% a nd 42.3%, res pective ly. A ccom plis h two cir culati ng pum ps repa ir and overha ul of four M ills, mainte nance teams to get exercis e.

Promote the work of energy savi ng and cons umption reducing, com plete the unit e nergy cons umptio n diag nosis, pl ant water bala nce test, 10 e nergy-savi ng proj ects .

Second, we s hould a dher e to "three" on t he economic benefits of improvi ng, outreach

8

Delegates, i nk the run-up t o the Spri ng Festival, wedheld one s essi on of four staff nce 2 013 -work shop, full back n 2012 , careful a nalysis the current sit uation, di scuss 20 13 dev elopment pla ns. Here, conse on be half of year my company 2 013 w orkity reports to posture the General Assembl y, for consi deration. Pillar I, 2012 ba inmillion 20 12, XX power compa nies a dher e to the 17 great pirit for guida comprehensively im plemmilli ent the entific conce pt of1.76 devel opment, promoti cost-lea ders hip strategy, focus on er imsupply plementati on, lea n manageme nt, continuously , smoot h pre sent safety situati power on of enterpri se manageme busine ss manag ement and scieperforma ntific andnce standardiz a nd the de dication of staff, a harmonious and democratic atm osphere of 0.512 the good Main i ndi cators are s follow s: -the battery indicator : power generati totale d 7.8 15 bill k Wh, t he annual budget implementati on capa city of forced 315 milli on k Wh, n increase of 757 million kWh. totaled billi on k Wh, exceedi ng sale s of 330 milli k Whathe annual Executive budget, ant n increase staff: of 729Hello! milli on Wh. --Security measures : unpla nne outages 2. 5 times. Norepresentatives persona l inj ury Confere accide nt occurred, no major accide nt iand above, no major fireof accidents wit hout environmental poll ution accide nts, s af ety for three cutive s to maintai n stabil to good . – Busines s financial i ndi cators: t otal profits of ck 255 Yua n, beyond the a nnual budget ofparty's 207 million Yuas n, beyond the nce, Datang company i ndex 4 1.89 on Ysci uan, an i ncreas e of mill ion Y uan, FCM ng as sessme nt at grade four. standards, --Energy: pow standard coal completi ng 312.25 g /k Wh,improve dow n 0.1 g/k Wh; integrated a uxiliary consumpti on rati o innt, 5. 12%, dow n 0.2 6%; polluta ntcontrol emissi ons greatly ed, reduced compare d to last year,manage car bon 0.09 g/k Wh, sulfur dioxide 0.10 4 g/k Wh NOx g/k situation. Wh; dust removal efficiency ofamore than 99 .8%. --Relia bi lity index: e quivale nton availability factor inion 93.47% , beyond increased 7.95% from a year e arlier. Equivalent outage ratea0.0 8%, 0. 16% reducti on over the s ame Sales peri od a year7.42 ear 5 lier. Major achieveme nts: first, we on s hould dhere to the two "manageme system" basis, stre ngthening technologi cal resear ch, stre ngthe n hi dde n hazar ds control and i ntrins ic safety Enter prise constructi on took ne w steps. -T he two "manageme nt system" for improvement. Focus on promoti ng the power of the com pany manageme nt system and the application a nd im pl ementation of the safety loop five-star manag ement system, improve t he safety management sy stem, realize the sy stem of safety control . Further reg ulate se curity routine s, safety supervisi on a nd manageme nt netw ork role to play to achieve cl ose d-l oop. Strengt heni ng the s upervision and manag ement of habit ual violati on of, strengtheni ng the s afety supervi sion of outsourcing contractors . Carrie d out i n spri ng and aut umn of se curity inspe ctions, flood contr ol and ins pe ction, safety production m onth, day supervisi on of production safety and the Olympi c Games a nd other

经济数学基础之线性代数
通过本节课的学习,掌握行列式的计算方法.

第 1 章 行列式

二、内容讲解

行列式的计算 行列式=按任何一行(列)展开 下面用具体例子说明.

a b 3 5 ? 3 ? 2 ? 5 ? (?1) ? 6 ? 5 ? 11 c d = ad ? bc ? 1 2
一个具体的行列式就是代表具体的一个数.再看一个三阶行列式.
1 3 ?1 0 2 4

5 ?7 0

可以按任何一行(列)展开

1?
按第一行展开=

2 4 3 4 3 2 ? 1? ? 0? ?7 0 5 0 5 ? 7 = 28 ? 20 ? 0 =8 3 2 1 ?1 1 ?1 ? 4? ? 0? 5 ?7 5 ?7 3 2 = 0 ? 4 ? (?7 ? 5) ? 0 =8

0?
按第三列展开=

注意:1.行列式计算一般按零元素较多的行(列)展开. 2.代数余子式的正负号是有规律的,一正一负相间隔.
a b a1 b1 d1 a b a 2 ? b2 c d c2 d2 b2 d2 c d c1 0 0 a2

问题:试证 答案左边=

0

0

c2

activities, compre hensive and tamping Safety Foundati on .. . Troubl eshooti ng, manageme nt mechanism, give full play to role of technical s upe rvision and realizati on of troubles hooting, ma nagement, impr oved process management. T his year completed t he boiler l ower hea der leakage, boi l er pres sure, a major ri sk management, completed 29 of great risks and 3 General problems of gover nance. Complete chemi stry lab constr ucti on, thermal control, a nd complete t he boiler s cale integrated ma nagement, host s hafting vi bration of 10 scie ntific and technologi cal pr ojects , such a s. 势 趋 来 未 础 基 为 此 及 以 , 程 过 展 发 善 改 步 逐 到 生 产 从 、 了 述 简 先 首 ; 机 动 电 流 直 磁 永 是 的 绍 介 要 主

Complete supercriti cal 630 MW on -line simulation system devel opme nt and applicati on of cir culati ng water MCC standby power transformation, t he transformation of de sulfuri zation w aste water, the unit water supply system of compre hensive treatment and dis charge valve modification of coal mill 5 key technologi cal transformation projects , group healt h is improve d. --S cience and technology i nnovati on is furt her increase d. Strengthen the characteristi cs of s uper critical unit major is sue s, Meanwhile, i nformation te chnol ogy achievements, t he company wa s named "China pow er information technol ogy be nchmarking

gradually cle aring the particularity of supercriti cal unit and regularity. Develope d motor oil time ma nagement, sw itch a ction times, statistics, coal-a ide d measureme nt software, impr oves the producti on lev el of lean management. I ncrea sed inve stment in s cie nce a nd technolog y, reporting s cie nce a nd technology proje cts and 14 techni cal pr oje ct total cost percentage of the tot al a nnual pr oduction output of 0.25% . "Large-s cale coal -fired power pla nt flue gas desulfur ization, de nitrification complete develo pment and applicati on of key technologie s" proje ct, won the national sci ence and technology pr ogress s econd prize. 6 30M W supercriti cal units optimized contr ol strategies and the 6 30M W devel opme nt and application of on-li ne sim ulation system for supercritica l unit s, supercriti cal 600 MW units of turbine drive n boiler feed pump set of compre hensive treatment of defect s Datang t echnol ogy respe ctively one or tw o and third.

enterprise s." --Repair and mainte nance ha s improved furt her. Modify the ins pe ction standards a nd sta ndar ds on a regular basis , standardizi ng work procedure s, checki ng and inspe ction proj ect. D eepe ning t he BFS++ system, and implements mai ntena nce i nformation share d. Reorgani zing RB l ogic again, and ensure the succe ss of the R B. Innovati ng the mecha ni sm of maintena nce ma nagement, impl emented a proje ct manager system. S uccessful completion of two aut onomous maintena nce, reliability improved steadil y. Impleme nting tw o c -level mainte nance, pr oject t hemselves 48.7% a nd 42.3%, res pective ly. A ccom plis h two cir culati ng pum ps repa ir and overha ul of four M ills, mainte nance teams to get exercis e.

Promote the work of energy savi ng and cons umption reducing, com plete the unit e nergy cons umption diag nosis, pl ant water bala nce test, 10 e nergy -savi ng proj ects .

Second, we s hould a dher e to "three" on t he economic benefits of improvi ng, outreach

——

9——

Delegates, staff: Hello! in t he run-up to t he Spri ng Festival, we hel d one se ssion of four staff represe ntatives Conference 201 3-w orkshop, full ba ck in 20 12, careful analysi s of the current s ituation, di scuss 2 013 development pla ns. Here, on be half of my company 2 013 work re ports t o the Ge neral Assembl y, for consi deration. Pillar I, 2012 back i n 201 2, XX power compani es adhere to t he party's 1 7 great spirit for gui dance, compre hensively impleme nt the s cientific conce pt of devel opme nt, promoti ng cost-lea ders hip strategy, standards, focus on im plementation, lea n manageme nt, conti nuously im prove, sm ooth prese nt safety situation of enter prise management, busi ness ma nagement a nd control scie ntific and standardi zed, a nd the de dicati on of staff, manage a harmonious and dem ocratic atmosphere of the g ood situati on. Main i ndi cators are as foll ows :-t he battery indi cator: power ge neration t otaled 7. 815 billion kWh, bey ond the a nnual budget impleme ntation capa city of 315 milli on k Wh, a n increase of 757 milli on k Wh. Sales totaled 7.425 billi on kWh, exceeding sales of 330 million kWh the a nnual Executive budget, an i ncrea se of 729 milli on k Wh. --Se curity meas ures : unplanne d outage s 2.5 times. No personal inj ury accident occurred, no major a cci de nt and a bove , no maj or fire acci de nts without envir onme ntal pol lution accidents, safety for three conse cutive years to mai ntain sta bility to good post ure. – Business financial indicator s: total pr ofits of 255 milli on Yua n, beyond the a nnual budget of 207 milli on Y uan, beyond the Datang compa ny index 41.89 milli on Y uan, a n increase of 1.76 milli on Yua n, FCM a sses sment at grade four. --Energy: power s upply standard coal compl eting 312 .25 g /kWh, dow n 0.1 g /kWh; integrate d auxiliary power consum ption ratio in 5. 12%, dow n 0.26% ; polluta nt emissions performance greatly reduced compared to last yea r, car bon 0.09 g/k Wh, sulfur dioxide 0.104 g/k Wh NOx 0.5 12 g/k Wh; dust removal efficiency of more tha n 99.8 %. --Reliability i ndex: equivalent availa bility factor in 93.47% , increa sed 7.95% from a year earlie r. Equivalent force d outage rate 0. 08% , 0.16% reducti on ov er the same period a year earlier. Major achieveme nts: first, we s hould a dhere to the tw o "management system" basi s, strengt heni ng technological research, stre ngthen hidden hazar ds control a nd i ntrinsi c safety Enterpri se constr ucti on took new steps . -T he two "manag ement system" for improvement. Focus on promoting t he power of the compa ny manageme nt system and t he applicati on a nd impl ementation of the safety loop five-star management system, impr ove the s afety management system, realize the system of safety control. Furt her regulate s ecur ity routine s, safety supervisi on a nd ma nagement net work role t o play to a chieve closed -loop. Stre ngthe ning the supervisi on a nd manageme nt of ha bitual vi olation of, strengthe ning the safety supervisi on of outsourci ng contra ctor s. Carrie d out in s pring a nd a utumn of se curity i ns pecti ons , flood contr ol and i ns pecti on, safety producti on month, day supervisi on of producti on safety and the Olympi c Games a nd ot her

d a (?1)
1?1

c1 a2 c2

d1 b2 ? c(?1) d2
2 ?1

b

a1

b1 b2 a ? ad (?1)1?1 2 d2 c2

0 0

0 a2 0 c2

b2 d2

? cb(?1)1?1

a2 c2

b2 d2

? (ad ? cb)

a2 c2

b2 d2

?

a b a2 c d c2

b2 d2
=右边

三、例题讲解
1 ?1 2 0 0 ?1 1 0 0 例 计算行列式 4 1 ? 2 0 3 2 0

分析:由性质 6 可知,行列式可以按任何一行(列)展开来求值.因为第二、 三行,第四列的零元素都较多,所以可选择其一展开,再进一步将其展成二阶行 列式,并计算结果. 解:按第三行展开

1 ?1 2 0 0 ?1 1 0 0 4 1 ?2
3 ? (?1) 2?3
=

0 ?1 2 0 1 ?1 2 3 3?1 3? 4 2 1 ? (?1) 5 ? 1 3 ? 2 ? (?1) 0 0 ? 1 1 ?2 0 4 1 ?2 0=
?1 2 1 ?1 ? 2 ? (?1) ? (?1) 2?3 1 ?2 4 1 == ? 3 ? (2 ? 2) ? 2(1 ? 4) ? ?10

四、课堂练习
a 1 0 ?1 b 1 0 ?1 c 0 ?1 练习 1 计算行列式 0 0 0 1 d
Complete supercriti cal 630 MW on-line simulation system devel opme nt and applicati on of cir culati ng water MCC standby power transformation, t he transformation of de sulfuri zation w aste water, the unit water supply system of compre hensive treatment and dis charge valve modification of coal mill 5 key technologi cal transformation projects , group healt h is improve d. --S cience and technology i nnovati on is furt her increase d. Strengthen the characteristi cs of s uper critical unit major is sue s, Meanwhile, i nformation te chnol ogy achievements, t he company wa s named "China pow er information technol ogy be nchmarking

activities, compre hensive and tamping Safety Foundati on .. . Troubl eshooti ng, manageme nt mechanism, give full play to role of technical s upe rvision and realizati on of troubles hooting, ma nagement, impr oved process management. T his year completed t he boiler l ower hea der leakage, boil er pres sure, a major ri sk management, completed 29 of gr eat risks and 3 General problems of gover nance. Complete chemi stry lab constr ucti on, thermal control, a nd complete t he boiler s cale integrated ma nagement, host s hafting vi bration of 10 scie ntific and technologi cal pr ojects , such a s. 势 趋 来 未 础 基 为 此 及 以 , 程 过 展 发 善 改 步 逐 到 生 产 从 、 了 述 简 先 首 ; 机 动 电 流 直 磁 永 是 的 绍 介 要 主

gradually cle aring the particularity of supercriti cal unit and regularity. Develope d motor oil time ma nagement, sw itch a ction times, statistics, coal-a ide d measureme nt software, impr oves the producti on lev el of lean management. I ncrea sed inve stment in s cie nce a nd technolog y, reporting s cie nce a nd technology proje cts and 14 techni cal pr oje ct total cost percentage of the total a nnual pr oduction output of 0.25% . "Large-s cale coal -fired power pla nt flue gas desulfur ization, de nitrification complete development and applicati on of key technologie s" proje ct , won the national sci ence and technology pr ogress s econd prize. 6 30M W supercriti cal units optimized contr ol strategies and the 6 30M W devel opme nt and application of on-li ne sim ulation system for supercritica l unit s, supercriti cal 600 MW units of turbine drive n bo iler feed pump set of compre hensive treatment of defect s Datang t echnol ogy respe ctively one or tw o and third.

enterprise s." --Repair and mainte nance ha s improved furt her. Modify the ins pe ction standards a nd sta ndar ds on a regular basis , standardizi ng work procedure s, checki ng and inspe ction proj ect. D eepe ning t he BFS++ system, and implements mai ntena nce i nformation share d. Reorgani zing RB l ogic again, and ensure the succe ss of the R B. Innovati ng the mechani sm of maintena nce ma nagement, impl emented a proje ct manager system. S uccessful completion of two aut onomous maintena nce, reliability improved steadil y. Impleme nting tw o c -level mainte nance, pr oject t hemselves 48.7% a nd 42.3%, res pective ly. A ccom plis h two cir culati ng pum ps repa ir and overha ul of four M ills, mainte nance teams to get exercis e.

Promote the work of energy savi ng and cons umption reducing, com plete the unit e nergy cons umptio n diag nosis, pl ant water bala nce test, 10 e nergy-savi ng proj ects .

Second, we s hould a dher e to "three" on t he economic benefits of improvi ng, outreach

10

Delegates, i nk the run-up t o the Spri ng Festival, wedheld one s essi on of four staff nce 2 013 -work shop, full back n 2012 , careful a nalysis the current sit uation, di scuss 20 13 dev elopment pla ns. Here, conse on be half of year my company 2 013 w orkity reports to posture the General Assembl y, for consi deration. Pillar I, 2012 ba inmillion 20 12, XX power compa nies a dher e to the 17 great pirit for guida comprehensively im plemmilli ent the entific conce pt of1.76 devel opment, promoti cost-lea ders hip strategy, focus on er imsupply plementati on, lea n manageme nt, continuously , smoot h pre sent safety situati power on of enterpri se manageme busine ss manag ement and scieperforma ntific andnce standardiz a nd the de dication of staff, a harmonious and democratic atm osphere of 0.512 the good Main i ndi cators are s follow s: -the battery indicator : power generati totale d 7.8 15 bill k Wh, t he annual budget implementati on capa city of forced 315 milli on k Wh, n increase of 757 million kWh. totaled billi on k Wh, exceedi ng sale s of 330 milli k Whathe annual Executive budget, ant n increase staff: of 729Hello! milli on Wh. --Security measures : unpla nne outages 2. 5 times. Norepresentatives persona l inj ury Confere accide nt occurred, no major accide nt iand above, no major fireof accidents wit hout environmental poll ution accide nts, s af ety for three cutive s to maintai n stabil to good . – Busines s financial i ndi cators: t otal profits of ck 255 Yua n, beyond the a nnual budget ofparty's 207 million Yuas n, beyond the nce, Datang company i ndex 4 1.89 on Ysci uan, an i ncreas e of mill ion Y uan, FCM ng as sessme nt at grade four. standards, --Energy: pow standard coal completi ng 312.25 g /k Wh,improve dow n 0.1 g/k Wh; integrated a uxiliary consumpti on rati o innt, 5. 12%, dow n 0.2 6%; polluta ntcontrol emissi ons greatly ed, reduced compare d to last year,manage car bon 0.09 g/k Wh, sulfur dioxide 0.10 4 g/k Wh NOx g/k situation. Wh; dust removal efficiency ofamore than 99 .8%. --Relia bi lity index: e quivale nton availability factor inion 93.47% , beyond increased 7.95% from a year e arlier. Equivalent outage ratea0.0 8%, 0. 16% reducti on over the s ame Sales peri od a year7.42 ear 5 lier. Major achieveme nts: first, we on s hould dhere to the two "manageme system" basis, stre ngthening technologi cal resear ch, stre ngthe n hi dde n hazar ds control and i ntrins ic safety Enter prise constructi on took ne w steps. -T he two "manageme nt system" for improvement. Focus on promoti ng the power of the com pany manageme nt system and the application a nd im pl ementation of the safety loop five-star manag ement system, improve t he safety management sy stem, realize the sy stem of safety control . Further reg ulate se curity routine s, safety supervisi on a nd manageme nt netw ork role to play to achieve cl ose d-l oop. Strengt heni ng the s upervision and manag ement of habit ual violati on of, strengtheni ng the s afety supervi sion of outsourcing contractors . Carrie d out i n spri ng and aut umn of se curity inspe ctions, flood contr ol and ins pe ction, safety production m onth, day supervisi on of production safety and the Olympi c Games a nd other

经济数学基础之线性代数

第 1 章 行列式

根据定义,按第一行展开,使其成为两个三阶行列式之和. 因为行列式第一行有较多的零元素,所以可采用“降阶法”,即先按第一行展 开,使其成为两个三阶行列式之和,然后再计算两个三阶行列式降阶,最后求出结

a 1 0 ?1 b 1 0 ?1 c 0 ?1 果. 0

0 0 1 d

b

1

0

?1

1

0

=

a ?1 c 1 ? 0 0 ?1 d 0

c 1 ?1 d

? 3 2 ? 2 ? 20 2 ?1 3 16 5 ?2 8 28 24 练习 2 计算行列式 4 ? 2 5
为了避免分数运算,先作变换“第一行加上第二行的 2 倍,即 ?+? 2;第三行加上第二行的 -2 倍,即 ?+?(-2);第四行加上第二行的-2 倍,即 ?+?(-2)” .

该行列式没有明显特点,采用哪种方法计算都可以,这里用“化三角行列式” 的方法进行计算.注意尽量避免分数运算.

? 3 2 ? 2 ? 20 2 ?1 3 16 ?+?2 5 ?2 8 28 ?+?(-2) ?+?(-2) 4 ?2 5 21

1 0 4 12 2 ? 1 3 16 1 0 2 ?4 0 0 ? 1 ? 11

五、课后作业
1.计算下列行列式:
?1 2 ? 2 1 4 4 ?1 8 ? 8

(1)

1 2 2 (2) 2

2 2 2 2

2 2 3 2

2 2 2 4

activities, compre hensive and tamping Safety Foundati on .. . Troubl eshooti ng, manageme nt mechanism, give full play to role of technical s upe rvision and realizati on of troubles hooting, ma nagement, impr oved process management. T his year completed t he boiler l ower hea der leakage, boi l er pres sure, a major ri sk management, completed 29 of great risks and 3 General problems of gover nance. Complete chemi stry lab constr ucti on, thermal control, a nd complete t he boiler s cale integrated ma nagement, host s hafting vi bration of 10 scie ntific and technologi cal pr ojects , such a s. 势 趋 来 未 础 基 为 此 及 以 , 程 过 展 发 善 改 步 逐 到 生 产 从 、 了 述 简 先 首 ; 机 动 电 流 直 磁 永 是 的 绍 介 要 主

Complete supercriti cal 630 MW on -line simulation system devel opme nt and applicati on of cir culati ng water MCC standby power transformation, t he transformation of de sulfuri zation w aste water, the unit water supply system of compre hensive treatment and dis charge valve modification of coal mill 5 key technologi cal transformation projects , group healt h is improve d. --S cience and technology i nnovati on is furt her increase d. Strengthen the characteristi cs of s uper critical unit major is sue s, Meanwhile, i nformation te chnol ogy achievements, t he company wa s named "China pow er information technol ogy be nchmarking

gradually cle aring the particularity of supercriti cal unit and regularity. Develope d motor oil time ma nagement, sw itch a ction times, statistics, coal-a ide d measureme nt software, impr oves the producti on lev el of lean management. I ncrea sed inve stment in s cie nce a nd technolog y, reporting s cie nce a nd technology proje cts and 14 techni cal pr oje ct total cost percentage of the tot al a nnual pr oduction output of 0.25% . "Large-s cale coal -fired power pla nt flue gas desulfur ization, de nitrification complete develo pment and applicati on of key technologie s" proje ct, won the national sci ence and technology pr ogress s econd prize. 6 30M W supercriti cal units optimized contr ol strategies and the 6 30M W devel opme nt and application of on-li ne sim ulation system for supercritica l unit s, supercriti cal 600 MW units of turbine drive n boiler feed pump set of compre hensive treatment of defect s Datang t echnol ogy respe ctively one or tw o and third.

enterprise s." --Repair and mainte nance ha s improved furt her. Modify the ins pe ction standards a nd sta ndar ds on a regular basis , standardizi ng work procedure s, checki ng and inspe ction proj ect. D eepe ning t he BFS++ system, and implements mai ntena nce i nformation share d. Reorgani zing RB l ogic again, and ensure the succe ss of the R B. Innovati ng the mecha ni sm of maintena nce ma nagement, impl emented a proje ct manager system. S uccessful completion of two aut onomous maintena nce, reliability improved steadil y. Impleme nting tw o c -level mainte nance, pr oject t hemselves 48.7% a nd 42.3%, res pective ly. A ccom plis h two cir culati ng pum ps repa ir and overha ul of four M ills, mainte nance teams to get exercis e.

Promote the work of energy savi ng and cons umption reducing, com plete the unit e nergy cons umption diag nosis, pl ant water bala nce test, 10 e nergy -savi ng proj ects .

Second, we s hould a dher e to "three" on t he economic benefits of improvi ng, outreach

——

11——

Delegates, staff: Hello! in t he run-up to t he Spri ng Festival, we hel d one se ssion of four staff represe ntatives Conference 201 3-w orkshop, full ba ck in 20 12, careful analysi s of the current s ituation, di scuss 2 013 development pla ns. Here, on be half of my company 2 013 work re ports t o the Ge neral Assembl y, for consi deration. Pillar I, 2012 back i n 201 2, XX power compani es adhere to t he party's 1 7 great spirit for gui dance, compre hensively impleme nt the s cientific conce pt of devel opme nt, promoti ng cost-lea ders hip strategy, standards, focus on im plementation, lea n manageme nt, conti nuously im prove, sm ooth prese nt safety situation of enter prise management, busi ness ma nagement a nd control scie ntific and standardi zed, a nd the de dicati on of staff, manage a harmonious and dem ocratic atmosphere of the g ood situati on. Main i ndi cators are as foll ows :-t he battery indi cator: power ge neration t otaled 7. 815 billion kWh, bey ond the a nnual budget impleme ntation capa city of 315 milli on k Wh, a n increase of 757 milli on k Wh. Sales totaled 7.425 billi on kWh, exceeding sales of 330 million kWh the a nnual Executive budget, an i ncrea se of 729 milli on k Wh. --Se curity meas ures : unplanne d outage s 2.5 times. No personal inj ury accident occurred, no major a cci de nt and a bove , no maj or fire acci de nts without envir onme ntal pol lution accidents, safety for three conse cutive years to mai ntain sta bility to good post ure. – Business financial indicator s: total pr ofits of 255 milli on Yua n, beyond the a nnual budget of 207 milli on Y uan, beyond the Datang compa ny index 41.89 milli on Y uan, a n increase of 1.76 milli on Yua n, FCM a sses sment at grade four. --Energy: power s upply standard coal compl eting 312 .25 g /kWh, dow n 0.1 g /kWh; integrate d auxiliary power consum ption ratio in 5. 12%, dow n 0.26% ; polluta nt emissions performance greatly reduced compared to last yea r, car bon 0.09 g/k Wh, sulfur dioxide 0.104 g/k Wh NOx 0.5 12 g/k Wh; dust removal efficiency of more tha n 99.8 %. --Reliability i ndex: equivalent availa bility factor in 93.47% , increa sed 7.95% from a year earlie r. Equivalent force d outage rate 0. 08% , 0.16% reducti on ov er the same period a year earlier. Major achieveme nts: first, we s hould a dhere to the tw o "management system" basi s, strengt heni ng technological research, stre ngthen hidden hazar ds control a nd i ntrinsi c safety Enterpri se constr ucti on took new steps . -T he two "manag ement system" for improvement. Focus on promoting t he power of the compa ny manageme nt system and t he applicati on a nd impl ementation of the safety loop five-star management system, impr ove the s afety management system, realize the system of safety control. Furt her regulate s ecur ity routine s, safety supervisi on a nd ma nagement net work role t o play to a chieve closed -loop. Stre ngthe ning the supervisi on a nd manageme nt of ha bitual vi olation of, strengthe ning the safety supervisi on of outsourci ng contra ctor s. Carrie d out in s pring a nd a utumn of se curity i ns pecti ons , flood contr ol and i ns pecti on, safety producti on month, day supervisi on of producti on safety and the Olympi c Games a nd ot her

4

21 3 5 0 6 1

4 2

1 1 0 (3) 1

2 3 1 2

0 5 5 3

1 6 6 4

36 1

13 1 3 5 0 0 7 3 0 0 0 0

x a a x ? a ? ? ? ? 2.计算 n 阶行列式 a a ? x
1. (1)48 (2)4 (3)-3 (4)-340

(4) ? a

3 2

n?1 2. ( x ? a) [(n ? 1)a ? x]

第四单元
一、学习目标

克拉默法则

克拉默法则是行列式在解线性方程组中的一个应用, 通过本节课的学习, 要知 道克拉默法则求线性方程组解的条件,了解克拉默法则的结论.

二、内容讲解
克拉默法则 设 n 个未知数的线性方程组为 ? a11 x1 ? a12 x 2 ? ? ? a1n x n ? b1 ?a x ? a x ? ? ? a x ? b ? 21 1 22 2 2n n 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?a n1 x1 ? a n 2 x 2 ? ? ? a nn x n ? bn

(1)

activities, compre hensive and tamping Safety Foundati on .. . Troubl eshooti ng, manageme nt mechanism, give full play to role of technical s upe rvision and realizati on of troubles hooting, ma nagement, impr oved process management. T his year completed t he boiler l ower hea der leakage, boil er pres sure, a major ri sk management, completed 29 of gr eat risks and 3 General problems of gover nance. Complete chemi stry lab constr ucti on, thermal control, a nd complete t he boiler s cale integrated ma nagement, host s hafting vi bration of 10 scie ntific and technologi cal pr ojects , such a s. 势 趋 来 未 础 基 为 此 及 以 , 程 过 展 发 善 改 步 逐 到 生 产 从 、 了 述 简 先 首 ; 机 动 电 流 直 磁 永 是 的 绍 介 要 主

Complete supercriti cal 630 MW on-line simulation system devel opme nt and applicati on of cir culati ng water MCC standby power transformation, t he transformation of de sulfuri zation w aste water, the unit water supply system of compre hensive treatment and dis charge valve modification of coal mill 5 key technologi cal transformation projects , group healt h is improve d. --S cience and technology i nnovati on is furt her increase d. Strengthen the characteristi cs of s uper critical unit major is sue s, Meanwhile, i nformation te chnol ogy achievements, t he company wa s named "China pow er information technol ogy be nchmarking

gradually cle aring the particularity of supercriti cal unit and regularity. Develope d motor oil time ma nagement, sw itch a ction times, statistics, coal-a ide d measureme nt software, impr oves the producti on lev el of lean management. I ncrea sed inve stment in s cie nce a nd technolog y, reporting s cie nce a nd technology proje cts and 14 techni cal pr oje ct total cost percentage of the total a nnual pr oduction output of 0.25% . "Large-s cale coal -fired power pla nt flue gas desulfur ization, de nitrification complete development and applicati on of key technologie s" proje ct , won the national sci ence and technology pr ogress s econd prize. 6 30M W supercriti cal units optimized contr ol strategies and the 6 30M W devel opme nt and application of on-li ne sim ulation system for supercritica l unit s, supercriti cal 600 MW units of turbine drive n bo iler feed pump set of compre hensive treatment of defect s Datang t echnol ogy respe ctively one or tw o and third.

enterprise s." --Repair and mainte nance ha s improved furt her. Modify the ins pe ction standards a nd sta ndar ds on a regular basis , standardizi ng work procedure s, checki ng and inspe ction proj ect. D eepe ning t he BFS++ system, and implements mai ntena nce i nformation share d. Reorgani zing RB l ogic again, and ensure the succe ss of the R B. Innovati ng the mechani sm of maintena nce ma nagement, impl emented a proje ct manager system. S uccessful completion of two aut onomous maintena nce, reliability improved steadil y. Impleme nting tw o c -level mainte nance, pr oject t hemselves 48.7% a nd 42.3%, res pective ly. A ccom plis h two cir culati ng pum ps repa ir and overha ul of four M ills, mainte nance teams to get exercis e.

Promote the work of energy savi ng and cons umption reducing, com plete the unit e nergy cons umptio n diag nosis, pl ant water bala nce test, 10 e nergy-savi ng proj ects .

Second, we s hould a dher e to "three" on t he economic benefits of improvi ng, outreach

12

Delegates, i nk the run-up t o the Spri ng Festival, wedheld one s essi on of four staff nce 2 013 -work shop, full back n 2012 , careful a nalysis the current sit uation, di scuss 20 13 dev elopment pla ns. Here, conse on be half of year my company 2 013 w orkity reports to posture the General Assembl y, for consi deration. Pillar I, 2012 ba inmillion 20 12, XX power compa nies a dher e to the 17 great pirit for guida comprehensively im plemmilli ent the entific conce pt of1.76 devel opment, promoti cost-lea ders hip strategy, focus on er imsupply plementati on, lea n manageme nt, continuously , smoot h pre sent safety situati power on of enterpri se manageme busine ss manag ement and scieperforma ntific andnce standardiz a nd the de dication of staff, a harmonious and democratic atm osphere of 0.512 the good Main i ndi cators are s follow s: -the battery indicator : power generati totale d 7.8 15 bill k Wh, t he annual budget implementati on capa city of forced 315 milli on k Wh, n increase of 757 million kWh. totaled billi on k Wh, exceedi ng sale s of 330 milli k Whathe annual Executive budget, ant n increase staff: of 729Hello! milli on Wh. --Security measures : unpla nne outages 2. 5 times. Norepresentatives persona l inj ury Confere accide nt occurred, no major accide nt iand above, no major fireof accidents wit hout environmental poll ution accide nts, s af ety for three cutive s to maintai n stabil to good . – Busines s financial i ndi cators: t otal profits of ck 255 Yua n, beyond the a nnual budget ofparty's 207 million Yuas n, beyond the nce, Datang company i ndex 4 1.89 on Ysci uan, an i ncreas e of mill ion Y uan, FCM ng as sessme nt at grade four. standards, --Energy: pow standard coal completi ng 312.25 g /k Wh,improve dow n 0.1 g/k Wh; integrated a uxiliary consumpti on rati o innt, 5. 12%, dow n 0.2 6%; polluta ntcontrol emissi ons greatly ed, reduced compare d to last year,manage car bon 0.09 g/k Wh, sulfur dioxide 0.10 4 g/k Wh NOx g/k situation. Wh; dust removal efficiency ofamore than 99 .8%. --Relia bi lity index: e quivale nton availability factor inion 93.47% , beyond increased 7.95% from a year e arlier. Equivalent outage ratea0.0 8%, 0. 16% reducti on over the s ame Sales peri od a year7.42 ear 5 lier. Major achieveme nts: first, we on s hould dhere to the two "manageme system" basis, stre ngthening technologi cal resear ch, stre ngthe n hi dde n hazar ds control and i ntrins ic safety Enter prise constructi on took ne w steps. -T he two "manageme nt system" for improvement. Focus on promoti ng the power of the com pany manageme nt system and the application a nd im pl ementation of the safety loop five-star manag ement system, improve t he safety management sy stem, realize the sy stem of safety control . Further reg ulate se curity routine s, safety supervisi on a nd manageme nt netw ork role to play to achieve cl ose d-l oop. Strengt heni ng the s upervision and manag ement of habit ual violati on of, strengtheni ng the s afety supervi sion of outsourcing contractors . Carrie d out i n spri ng and aut umn of se curity inspe ctions, flood contr ol and ins pe ction, safety production m onth, day supervisi on of production safety and the Olympi c Games a nd other

经济数学基础之线性代数
a11 D? a 21 ? a n1 a12 a 22 ? ? a1n ? a2n ? ?

第 1 章 行列式

记行列式 的元素

a n 2 ? a nn

称为方程组 (1) 的系数行列式. 将 D 中第 j 列
Dj

a1 j , a2 j ,?, anj

分别换成常数 b1 ,b2 ,? ,bn 而得到的行列式记作



克拉默法则

如果线性方程组(1)的系数行列式 D ? 0 ,那
x1 ? D D1 D , x 2 ? 2 , ?, x n ? n D D D

么它有惟一解

(2)

证将(2)式分别代入方程组(1)的第 i 个方程的左端的 x1 , x2 ,?, xn 中,有
ai1 D D1 D ? ai 2 2 ? ? ? ain n D D D
Dj

(3)

将 (3) 中的

D 按第 j 列展开, 再注意到 j 中第 j 列元素的代数余子式和 D 中

A 第 j 列元素的代数余子式 ij 是相同的, 因此有

D j ? b1 A1 j ? b2 A2 j ? ? ? bn Anj
ai1

( j ? 1,2,?, n)

(4)

把(4)代入(3),有
?

D D1 D ? ai 2 2 ? ? ? ain n D D D

1 ?ai1 ?b1 A11 ? b2 A21 ? ?bi Ai1 ? ?bn An1 ? ? a ?b A ? b A ? ?b A ? ?b A ? D i 2 1 12 2 22 i i2 n n2

…+… ? ain ?b1 A1n ? b2 A2n ? ?bi Ain ? ?bn Ann ?? 把小括弧打开重新组合得

activities, compre hensive and tamping Safety Foundati on .. . Troubl eshooti ng, manageme nt mechanism, give full play to role of technical s upe rvision and realizati on of troubles hooting, ma nagement, impr oved process management. T his year completed t he boiler l ower hea der leakage, boi l er pres sure, a major ri sk management, completed 29 of great risks and 3 General problems of gover nance. Complete chemi stry lab constr ucti on, thermal control, a nd complete t he boiler s cale integrated ma nagement, host s hafting vi bration of 10 scie ntific and technologi cal pr ojects , such a s. 势 趋 来 未 础 基 为 此 及 以 , 程 过 展 发 善 改 步 逐 到 生 产 从 、 了 述 简 先 首 ; 机 动 电 流 直 磁 永 是 的 绍 介 要 主

Complete supercriti cal 630 MW on -line simulation system devel opme nt and applicati on of cir culati ng water MCC standby power transformation, t he transformation of de sulfuri zation w aste water, the unit water supply system of compre hensive treatment and dis charge valve modification of coal mill 5 key technologi cal transformation projects , group healt h is improve d. --S cience and technology i nnovati on is furt her increase d. Strengthen the characteristi cs of s uper critical unit major is sue s, Meanwhile, i nformation te chnol ogy achievements, t he company wa s named "China pow er information technol ogy be nchmarking

gradually cle aring the particularity of supercriti cal unit and regularity. Develope d motor oil time ma nagement, sw itch a ction times, statistics, coal-a ide d measureme nt software, impr oves the producti on lev el of lean management. I ncrea sed inve stment in s cie nce a nd technolog y, reporting s cie nce a nd technology proje cts and 14 techni cal pr oje ct total cost percentage of the tot al a nnual pr oduction output of 0.25% . "Large-s cale coal -fired power pla nt flue gas desulfur ization, de nitrification complete develo pment and applicati on of key technologie s" proje ct, won the national sci ence and technology pr ogress s econd prize. 6 30M W supercriti cal units optimized contr ol strategies and the 6 30M W devel opme nt and application of on-li ne sim ulation system for supercritica l unit s, supercriti cal 600 MW units of turbine drive n boiler feed pump set of compre hensive treatment of defect s Datang t echnol ogy respe ctively one or tw o and third.

enterprise s." --Repair and mainte nance ha s improved furt her. Modify the ins pe ction standards a nd sta ndar ds on a regular basis , standardizi ng work procedure s, checki ng and inspe ction proj ect. D eepe ning t he BFS++ system, and implements mai ntena nce i nformation share d. Reorgani zing RB l ogic again, and ensure the succe ss of the R B. Innovati ng the mecha ni sm of maintena nce ma nagement, impl emented a proje ct manager system. S uccessful completion of two aut onomous maintena nce, reliability improved steadil y. Impleme nting tw o c -level mainte nance, pr oject t hemselves 48.7% a nd 42.3%, res pective ly. A ccom plis h two cir culati ng pum ps repa ir and overha ul of four M ills, mainte nance teams to get exercis e.

Promote the work of energy savi ng and cons umption reducing, com plete the unit e nergy cons umption diag nosis, pl ant water bala nce test, 10 e nergy -savi ng proj ects .

Second, we s hould a dher e to "three" on t he economic benefits of improvi ng, outreach

——

13——

Delegates, staff: Hello! in t he run-up to t he Spri ng Festival, we hel d one se ssion of four staff represe ntatives Conference 201 3-w orkshop, full ba ck in 20 12, careful analysi s of the current s ituation, di scuss 2 013 development pla ns. Here, on be half of my company 2 013 work re ports t o the Ge neral Assembl y, for consi deration. Pillar I, 2012 back i n 201 2, XX power compani es adhere to t he party's 1 7 great spirit for gui dance, compre hensively impleme nt the s cientific conce pt of devel opme nt, promoti ng cost-lea ders hip strategy, standards, focus on im plementation, lea n manageme nt, conti nuously im prove, sm ooth prese nt safety situation of enter prise management, busi ness ma nagement a nd control scie ntific and standardi zed, a nd the de dicati on of staff, manage a harmonious and dem ocratic atmosphere of the g ood situati on. Main i ndi cators are as foll ows :-t he battery indi cator: power ge neration t otaled 7. 815 billion kWh, bey ond the a nnual budget impleme ntation capa city of 315 milli on k Wh, a n increase of 757 milli on k Wh. Sales totaled 7.425 billi on kWh, exceeding sales of 330 million kWh the a nnual Executive budget, an i ncrea se of 729 milli on k Wh. --Se curity meas ures : unplanne d outage s 2.5 times. No personal inj ury accident occurred, no major a cci de nt and a bove , no maj or fire acci de nts without envir onme ntal pol lution accidents, safety for three conse cutive years to mai ntain sta bility to good post ure. – Business financial indicator s: total pr ofits of 255 milli on Yua n, beyond the a nnual budget of 207 milli on Y uan, beyond the Datang compa ny index 41.89 milli on Y uan, a n increase of 1.76 milli on Yua n, FCM a sses sment at grade four. --Energy: power s upply standard coal compl eting 312 .25 g /kWh, dow n 0.1 g /kWh; integrate d auxiliary power consum ption ratio in 5. 12%, dow n 0.26% ; polluta nt emissions performance greatly reduced compared to last yea r, car bon 0.09 g/k Wh, sulfur dioxide 0.104 g/k Wh NOx 0.5 12 g/k Wh; dust removal efficiency of more tha n 99.8 %. --Reliability i ndex: equivalent availa bility factor in 93.47% , increa sed 7.95% from a year earlie r. Equivalent force d outage rate 0. 08% , 0.16% reducti on ov er the same period a year earlier. Major achieveme nts: first, we s hould a dhere to the tw o "management system" basi s, strengt heni ng technological research, stre ngthen hidden hazar ds control a nd i ntrinsi c safety Enterpri se constr ucti on took new steps . -T he two "manag ement system" for improvement. Focus on promoting t he power of the compa ny manageme nt system and t he applicati on a nd impl ementation of the safety loop five-star management system, impr ove the s afety management system, realize the system of safety control. Furt her regulate s ecur ity routine s, safety supervisi on a nd ma nagement net work role t o play to a chieve closed -loop. Stre ngthe ning the supervisi on a nd manageme nt of ha bitual vi olation of, strengthe ning the safety supervisi on of outsourci ng contra ctor s. Carrie d out in s pring a nd a utumn of se curity i ns pecti ons , flood contr ol and i ns pecti on, safety producti on month, day supervisi on of producti on safety and the Olympi c Games a nd ot her

?

1 ?b1 ?ai1 A11 ? ai 2 A12 ? ? ? ain A1n ? D ? b2 ?ai1 A21 ? ai 2 A22 ? ? ? ain A2 n ? ?? ? ? ? bi ?ai1 Ai1 ? ai 2 Ai 2 ? ? ? ain Ain ? ?? ? ? ? bn ?ai1 An1 ? ai 2 An 2 ? ? ? ain Ann ?? ? bi

因由性质 6 和性质 7

?0 ai1 Ak1 ? ai 2 Ak 2 ? ? ? ain Akn ? ? ?D
ai1 D D1 D ? ai 2 2 ? ? ? ain n ? bi D D D

i?k i ? k 故上式等于 bi ,即

下面再证明方程组(1)的解是惟一的.设

x1 ? c1 , x2 ? c2 ,?, xn ? cn
为方程组(1)的任意一组解.于是 ? a11c1 ? a12 c 2 ? ? ? a1n c n ? b1 ?a c ? a c ? ? ? a c ? b ? 21 1 22 2 2n n 2 ? ? ???????????? ? A A A ?a n1c1 ? a n 2 c 2 ? ? ? a nn c n ? bn (5)用 1 j , 2 j ,… n j 分别乘以(5)式 的第一、第二、…、第 n 个等式,再把 n 个等式两边相加,得

(a11 A1 j ? a21 A2 j ? ? ? an1 Anj )c1 ? ? ? (a1 j A1 j ? a2 j A2 j ? ? ? anj Anj )c j ? ? ? (a1n A1 j ? a2n A2 j ? ? ? ann Anj )cn ? b1 A1 j ? b2 A2 j ? ? ? bn Anj
根据性质 6 和性质 7,上式即为 因为 D ? 0 ,所以 推论 1
cj ? Dj D

Dc j ? D j ( j ? 1,2,?, n)

( j ? 1,2, ?, n)

克拉默法则有以下两个推论:

如果齐次线性方程组的系数行列式 D ? 0 , 那么

它只有零解.
activities, compre hensive and tamping Safety Foundati on .. . Troubl eshooti ng, manageme nt mechanism, give full play to role of technical s upe rvision and realizati on of troubles hooting, ma nagement, impr oved process management. T his year completed t he boiler l ower hea der leakage, boil er pres sure, a major ri sk management, completed 29 of gr eat risks and 3 General problems of gover nance. Complete chemi stry lab constr ucti on, thermal control, a nd complete t he boiler s cale integrated ma nagement, host s hafting vi bration of 10 scie ntific and technologi cal pr ojects , such a s. 势 趋 来 未 础 基 为 此 及 以 , 程 过 展 发 善 改 步 逐 到 生 产 从 、 了 述 简 先 首 ; 机 动 电 流 直 磁 永 是 的 绍 介 要 主 Complete supercriti cal 630 MW on-line simulation system devel opme nt and applicati on of cir culati ng water MCC standby power transformation, t he transformation of de sulfuri zation w aste water, the unit water supply system of compre hensive treatment and dis charge valve modification of coal mill 5 key technologi cal transformation projects , group healt h is improve d. --S cience and technology i nnovati on is furt her increase d. Strengthen the characteristi cs of s uper critical unit major is sue s, Meanwhile, i nformation te chnol ogy achievements, t he company wa s named "China pow er information technol ogy be nchmarking gradually cle aring the particularity of supercriti cal unit and regularity. Develope d motor oil time ma nagement, sw itch a ction times, statistics, coal-a ide d measureme nt software, impr oves the producti on lev el of lean management. I ncrea sed inve stment in s cie nce a nd technolog y, reporting s cie nce a nd technology proje cts and 14 techni cal pr oje ct total cost percentage of the total a nnual pr oduction output of 0.25% . "Large-s cale coal -fired power pla nt flue gas desulfur ization, de nitrification complete development and applicati on of key technologie s" proje ct , won the national sci ence and technology pr ogress s econd prize. 6 30M W supercriti cal units optimized contr ol strategies and the 6 30M W devel opme nt and application of on-li ne sim ulation system for supercritica l unit s, supercriti cal 600 MW units of turbine drive n bo iler feed pump set of compre hensive treatment of defect s Datang t echnol ogy respe ctively one or tw o and third.

enterprise s." --Repair and mainte nance ha s improved furt her. Modify the ins pe ction standards a nd sta ndar ds on a regular basis , standardizi ng work procedure s, checki ng and inspe ction proj ect. D eepe ning t he BFS++ system, and implements mai ntena nce i nformation share d. Reorgani zing RB l ogic again, and ensure the succe ss of the R B. Innovati ng the mechani sm of maintena nce ma nagement, impl emented a proje ct manager system. S uccessful completion of two aut onomous maintena nce, reliability improved steadil y. Impleme nting tw o c -level mainte nance, pr oject t hemselves 48.7% a nd 42.3%, res pective ly. A ccom plis h two cir culati ng pum ps repa ir and overha ul of four M ills, mainte nance teams to get exercis e.

Promote the work of energy savi ng and cons umption reducing, com plete the unit e nergy cons umptio n diag nosis, pl ant water bala nce test, 10 e nergy-savi ng proj ects .

Second, we s hould a dher e to "three" on t he economic benefits of improvi ng, outreach

14

Delegates, i nk the run-up t o the Spri ng Festival, wedheld one s essi on of four staff nce 2 013 -work shop, full back n 2012 , careful a nalysis the current sit uation, di scuss 20 13 dev elopment pla ns. Here, conse on be half of year my company 2 013 w orkity reports to posture the General Assembl y, for consi deration. Pillar I, 2012 ba inmillion 20 12, XX power compa nies a dher e to the 17 great pirit for guida comprehensively im plemmilli ent the entific conce pt of1.76 devel opment, promoti cost-lea ders hip strategy, focus on er imsupply plementati on, lea n manageme nt, continuously , smoot h pre sent safety situati power on of enterpri se manageme busine ss manag ement and scieperforma ntific andnce standardiz a nd the de dication of staff, a harmonious and democratic atm osphere of 0.512 the good Main i ndi cators are s follow s: -the battery indicator : power generati totale d 7.8 15 bill k Wh, t he annual budget implementati on capa city of forced 315 milli on k Wh, n increase of 757 million kWh. totaled billi on k Wh, exceedi ng sale s of 330 milli k Whathe annual Executive budget, ant n increase staff: of 729Hello! milli on Wh. --Security measures : unpla nne outages 2. 5 times. Norepresentatives persona l inj ury Confere accide nt occurred, no major accide nt iand above, no major fireof accidents wit hout environmental poll ution accide nts, s af ety for three cutive s to maintai n stabil to good . – Busines s financial i ndi cators: t otal profits of ck 255 Yua n, beyond the a nnual budget ofparty's 207 million Yuas n, beyond the nce, Datang company i ndex 4 1.89 on Ysci uan, an i ncreas e of mill ion Y uan, FCM ng as sessme nt at grade four. standards, --Energy: pow standard coal completi ng 312.25 g /k Wh,improve dow n 0.1 g/k Wh; integrated a uxiliary consumpti on rati o innt, 5. 12%, dow n 0.2 6%; polluta ntcontrol emissi ons greatly ed, reduced compare d to last year,manage car bon 0.09 g/k Wh, sulfur dioxide 0.10 4 g/k Wh NOx g/k situation. Wh; dust removal efficiency ofamore than 99 .8%. --Relia bi lity index: e quivale nton availability factor inion 93.47% , beyond increased 7.95% from a year e arlier. Equivalent outage ratea0.0 8%, 0. 16% reducti on over the s ame Sales peri od a year7.42 ear 5 lier. Major achieveme nts: first, we on s hould dhere to the two "manageme system" basis, stre ngthening technologi cal resear ch, stre ngthe n hi dde n hazar ds control and i ntrins ic safety Enter prise constructi on took ne w steps. -T he two "manageme nt system" for improvement. Focus on promoti ng the power of the com pany manageme nt system and the application a nd im pl ementation of the safety loop five-star manag ement system, improve t he safety management sy stem, realize the sy stem of safety control . Further reg ulate se curity routine s, safety supervisi on a nd manageme nt netw ork role to play to achieve cl ose d-l oop. Strengt heni ng the s upervision and manag ement of habit ual violati on of, strengtheni ng the s afety supervi sion of outsourcing contractors . Carrie d out i n spri ng and aut umn of se curity inspe ctions, flood contr ol and ins pe ction, safety production m onth, day supervisi on of production safety and the Olympi c Games a nd other

经济数学基础之线性代数
推论 2 问题:对任一线性方程组都可用克拉默法则求解吗?

第 1 章 行列式

齐次线性方程组有非零解的必要条件是系数行列式 D ? 0 .

答案 不对.当线性方程组中的未知量个数与方程个数不一样;或未知量个数 与方程个数相同,但其系数行列式等于零时,不能使用克拉默法则.

三、例题讲解
?? 3x1 ? 4 x 2 ? 6 ? ?2 x1 ? 5 x 2 ? ?7

例 利用克拉默法则解下列方程组

分析: 这是一个两个变量、 两个方程的方程组, 它满足了克拉默法则一个条件. 克 拉默法则的另一个条件是要求系数行列式的值不等于零.因此,先求出方程组的系 数行列式的值,若它的值不等于零,说明该方程组有惟一解,然后求常数项替代后 的行列式的值,再用克拉默法则给出的公式求出解. 解:因为系数行列式

D?

?3 4 ? ?? 3? ? ?? 5? ? 4 ? 2 2 ?5 ? 15 ? 8 ? 7 ? 0 6 4 ?3 6 D D 2 9 ? ?2 D2 ? ?9 x1 ? 1 ? ? x2 ? 2 ? ?7 ?5 2 ? 7 D 7, D 7 , ,所以



D1 ?

四、课堂练习
k 取什么值时,下列方程组有唯一解?有唯一解时求出解.

? x1 ? x 2 ? kx3 ? 1 ? ?? x1 ? kx2 ? x3 ? ?1 ?x ? x ? 2x ? 0 2 3 ? 1

对行列式作变换“第二行加上第一行的 1 倍,即 ?+?;第三行加上第一行的-1 倍,即 ?+?(-1)”.
activities, compre hensive and tamping Safety Foundati on .. . Troubl eshooti ng, manageme nt mechanism, give full play to role of technical s upe rvision and realizati on of troubles hooting, ma nagement, impr oved process management. T his year completed t he boiler l ower hea der leakage, boi l er pres sure, a major ri sk management, completed 29 of great risks and 3 General problems of gover nance. Complete chemi stry lab constr ucti on, thermal control, a nd complete t he boiler s cale integrated ma nagement, host s hafting vi bration of 10 scie ntific and technologi cal pr ojects , such a s. 势 趋 来 未 础 基 为 此 及 以 , 程 过 展 发 善 改 步 逐 到 生 产 从 、 了 述 简 先 首 ; 机 动 电 流 直 磁 永 是 的 绍 介 要 主 Complete supercriti cal 630 MW on -line simulation system devel opme nt and applicati on of cir culati ng water MCC standby power transformation, t he transformation of de sulfuri zation w aste water, the unit water supply system of compre hensive treatment and dis charge valve modification of coal mill 5 key technologi cal transformation projects , group healt h is improve d. --S cience and technology i nnovati on is furt her increase d. Strengthen the characteristi cs of s uper critical unit major is sue s, Meanwhile, i nformation te chnol ogy achievements, t he company wa s named "China pow er information technol ogy be nchmarking gradually cle aring the particularity of supercriti cal unit and regularity. Develope d motor oil time ma nagement, sw itch a ction times, statistics, coal-a ide d measureme nt software, impr oves the producti on lev el of lean management. I ncrea sed inve stment in s cie nce a nd technolog y, reporting s cie nce a nd technology proje cts and 14 techni cal pr oje ct total cost percentage of the tot al a nnual pr oduction output of 0.25% . "Large-s cale coal -fired power pla nt flue gas desulfur ization, de nitrification complete develo pment and applicati on of key technologie s" proje ct, won the national sci ence and technology pr ogress s econd prize. 6 30M W supercriti cal units optimized contr ol strategies and the 6 30M W devel opme nt and application of on-li ne sim ulation system for supercritica l unit s, supercriti cal 600 MW units of turbine drive n boiler feed pump set of compre hensive treatment of defect s Datang t echnol ogy respe ctively one or tw o and third.

enterprise s." --Repair and mainte nance ha s improved furt her. Modify the ins pe ction standards a nd sta ndar ds on a regular basis , standardizi ng work procedure s, checki ng and inspe ction proj ect. D eepe ning t he BFS++ system, and implements mai ntena nce i nformation share d. Reorgani zing RB l ogic again, and ensure the succe ss of the R B. Innovati ng the mecha ni sm of maintena nce ma nagement, impl emented a proje ct manager system. S uccessful completion of two aut onomous maintena nce, reliability improved steadil y. Impleme nting tw o c -level mainte nance, pr oject t hemselves 48.7% a nd 42.3%, res pective ly. A ccom plis h two cir culati ng pum ps repa ir and overha ul of four M ills, mainte nance teams to get exercis e.

Promote the work of energy savi ng and cons umption reducing, com plete the unit e nergy cons umption diag nosis, pl ant water bala nce test, 10 e nergy -savi ng proj ects .

Second, we s hould a dher e to "three" on t he economic benefits of improvi ng, outreach

——

15——

Delegates, staff: Hello! in t he run-up to t he Spri ng Festival, we hel d one se ssion of four staff represe ntatives Conference 201 3-w orkshop, full ba ck in 20 12, careful analysi s of the current s ituation, di scuss 2 013 development pla ns. Here, on be half of my company 2 013 work re ports t o the Ge neral Assembl y, for consi deration. Pillar I, 2012 back i n 201 2, XX power compani es adhere to t he party's 1 7 great spirit for gui dance, compre hensively impleme nt the s cientific conce pt of devel opme nt, promoti ng cost-lea ders hip strategy, standards, focus on im plementation, lea n manageme nt, conti nuously im prove, sm ooth prese nt safety situation of enter prise management, busi ness ma nagement a nd control scie ntific and standardi zed, a nd the de dicati on of staff, manage a harmonious and dem ocratic atmosphere of the g ood situati on. Main i ndi cators are as foll ows :-t he battery indi cator: power ge neration t otaled 7. 815 billion kWh, bey ond the a nnual budget impleme ntation capa city of 315 milli on k Wh, a n increase of 757 milli on k Wh. Sales totaled 7.425 billi on kWh, exceeding sales of 330 million kWh the a nnual Executive budget, an i ncrea se of 729 milli on k Wh. --Se curity meas ures : unplanne d outage s 2.5 times. No personal inj ury accident occurred, no major a cci de nt and a bove , no maj or fire acci de nts without envir onme ntal pol lution accidents, safety for three conse cutive years to mai ntain sta bility to good post ure. – Business financial indicator s: total pr ofits of 255 milli on Yua n, beyond the a nnual budget of 207 milli on Y uan, beyond the Datang compa ny index 41.89 milli on Y uan, a n increase of 1.76 milli on Yua n, FCM a sses sment at grade four. --Energy: power s upply standard coal compl eting 312 .25 g /kWh, dow n 0.1 g /kWh; integrate d auxiliary power consum ption ratio in 5. 12%, dow n 0.26% ; polluta nt emissions performance greatly reduced compared to last yea r, car bon 0.09 g/k Wh, sulfur dioxide 0.104 g/k Wh NOx 0.5 12 g/k Wh; dust removal efficiency of more tha n 99.8 %. --Reliability i ndex: equivalent availa bility factor in 93.47% , increa sed 7.95% from a year earlie r. Equivalent force d outage rate 0. 08% , 0.16% reducti on ov er the same period a year earlier. Major achieveme nts: first, we s hould a dhere to the tw o "management system" basi s, strengt heni ng technological research, stre ngthen hidden hazar ds control a nd i ntrinsi c safety Enterpri se constr ucti on took new steps . -T he two "manag ement system" for improvement. Focus on promoting t he power of the compa ny manageme nt system and t he applicati on a nd impl ementation of the safety loop five-star management system, impr ove the s afety management system, realize the system of safety control. Furt her regulate s ecur ity routine s, safety supervisi on a nd ma nagement net work role t o play to a chieve closed -loop. Stre ngthe ning the supervisi on a nd manageme nt of ha bitual vi olation of, strengthe ning the safety supervisi on of outsourci ng contra ctor s. Carrie d out in s pring a nd a utumn of se curity i ns pecti ons , flood contr ol and i ns pecti on, safety producti on month, day supervisi on of producti on safety and the Olympi c Games a nd ot her

这是三个未知量三个方程的线性方程组,由克拉默法则知,当系数行列式 D ?0 时,方程 组有唯一解.所以,先求系数行列式的值.

1

1

k

1

1

k

D ? ?1 k 1 1 ?1 2

0 k ?1 k ?1 0 ?2 2?k

五、课后作业
用克莱姆法则解下列方程组
?2 ?2 x1 ? x 2 ? ? x1 ? x 2 ? 4 x3 ? 1 ? x ? 2x ? 1 2 3 ?
x3 ? ?

1.

2.

? x1 ? x 2 ? x3 ? x 4 ? ?7 ? x ? 3x ? x ? 8 ? 1 3 4 ? ? x1 ? 2 x 2 ? x3 ? x 4 ? ?2 ? ?2 x1 ? 2 x 2 ? 2 x3 ? x 4 ? ?4

1. x1 ? 3 , x2 ? 4 ,

3 35 10 x2 ? x3 ? x ? ? 2 2 ,2. 1 3 , 2 , x4 ? ?20 ,

activities, compre hensive and tamping Safety Foundati on .. . Troubl eshooti ng, manageme nt mechanism, give full play to role of technical s upe rvision and realizati on of troubles hooting, ma nagement, impr oved process management. T his year completed t he boiler l ower hea der leakage, boil er pres sure, a major ri sk management, completed 29 of gr eat risks and 3 General problems of gover nance. Complete chemi stry lab constr ucti on, thermal control, a nd complete t he boiler s cale integrated ma nagement, host s hafting vi bration of 10 scie ntific and technologi cal pr ojects , such a s. 势 趋 来 未 础 基 为 此 及 以 , 程 过 展 发 善 改 步 逐 到 生 产 从 、 了 述 简 先 首 ; 机 动 电 流 直 磁 永 是 的 绍 介 要 主

Complete supercriti cal 630 MW on-line simulation system devel opme nt and applicati on of cir culati ng water MCC standby power transformation, t he transformation of de sulfuri zation w aste water, the unit water supply system of compre hensive treatment and dis charge valve modification of coal mill 5 key technologi cal transformation projects , group healt h is improve d. --S cience and technology i nnovati on is furt her increase d. Strengthen the characteristi cs of s uper critical unit major is sue s, Meanwhile, i nformation te chnol ogy achievements, t he company wa s named "China pow er information technol ogy be nchmarking

gradually cle aring the particularity of supercriti cal unit and regularity. Develope d motor oil time ma nagement, sw itch a ction times, statistics, coal-a ide d measureme nt software, impr oves the producti on lev el of lean management. I ncrea sed inve stment in s cie nce a nd technolog y, reporting s cie nce a nd technology proje cts and 14 techni cal pr oje ct total cost percentage of the total a nnual pr oduction output of 0.25% . "Large-s cale coal -fired power pla nt flue gas desulfur ization, de nitrification complete development and applicati on of key technologie s" proje ct , won the national sci ence and technology pr ogress s econd prize. 6 30M W supercriti cal units optimized contr ol strategies and the 6 30M W devel opme nt and application of on-li ne sim ulation system for supercritica l unit s, supercriti cal 600 MW units of turbine drive n bo iler feed pump set of compre hensive treatment of defect s Datang t echnol ogy respe ctively one or tw o and third.

enterprise s." --Repair and mainte nance ha s improved furt her. Modify the ins pe ction standards a nd sta ndar ds on a regular basis , standardizi ng work procedure s, checki ng and inspe ction proj ect. D eepe ning t he BFS++ system, and implements mai ntena nce i nformation share d. Reorgani zing RB l ogic again, and ensure the succe ss of the R B. Innovati ng the mechani sm of maintena nce ma nagement, impl emented a proje ct manager system. S uccessful completion of two aut onomous maintena nce, reliability improved steadil y. Impleme nting tw o c -level mainte nance, pr oject t hemselves 48.7% a nd 42.3%, res pective ly. A ccom plis h two cir culati ng pum ps repa ir and overha ul of four M ills, mainte nance teams to get exercis e.

Promote the work of energy savi ng and cons umption reducing, com plete the unit e nergy cons umptio n diag nosis, pl ant water bala nce test, 10 e nergy-savi ng proj ects .

Second, we s hould a dher e to "three" on t he economic benefits of improvi ng, outreach

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