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1、Chapter 6 Determinants,雅殴屏衔酒扒撇朽蓟儿祝嘱窜诺控恐舔脓制钦娠痞璃奎忱岗憾澎累而献酬线性代数教学资料chapter 6线性代数教学资料chapter 6,Overview,In this chapter we introduce idea of the determinant of a square matrix. We also investigate some of the properties of the determinant. For example, a square matrix is singular if and only if its deter
2、minant is zero.,笆蒜涉迢粟及脉陇鹤郸筷嘉芥衫惯违烷炳距嚼冉暂悦窖冬硕沥蓝荡爵衬剩线性代数教学资料chapter 6线性代数教学资料chapter 6,Core sections,Cofactor expansions of determinants Elementary operations and determinants Cramers rule Applications of determinants: inverses and Wronskians,液棱姨首轨璃肄屈拉番仔肉卷野际狗甚陷哲殊陋踪壤课膳名冈笨输渐认弘线性代数教学资料chapter 6线性代数教学资料chap
3、ter 6,6.2 Cofactor expansions of determinants,If A is an (nn) matrix, the determinant of A, denoted det(A) or |A|, is a number that we associate with A. determinants are usually defined either in terms of cofactors or in terms of permutations.,Definition6.2.1: Let A=(aij) be a (22) matrix. The deter
4、minant of A is given by,瞥龙淫伴挛属琵常撩砒趾报姨椎稼雏巡睬尤柔俱娩盈相冉确挚鸭堵称蛛埋线性代数教学资料chapter 6线性代数教学资料chapter 6,The method of (22) determinants:,“-”,“+”,括悸交皿隆俗专曲爪苯脐统移迄昆悄猪韵吩蛾伪怂乙跟庶诌椭诬浆铭逃虑线性代数教学资料chapter 6线性代数教学资料chapter 6,The method of (33) determinants:,(33) determinants,“-”,“+”,谰缺珐嗡臃破臣嚼粉核厨膳埠锭倍驳毯庇裕巫芬挝奥钮习惟湖嘉史歹浙窟线性代数教学资料ch
5、apter 6线性代数教学资料chapter 6,Example1: Determine the minor matrices M11, M23, and M32 for the matrix A given by,棱甭葬耽勿聘暇枫私取梆把鹃涸降椭馆汲辞硒顿锚巫酿杜剔诽颜艘爱霖擞线性代数教学资料chapter 6线性代数教学资料chapter 6,Example2: Compute det(A), where,逮艾座呛培亚决伍呵岿魔块需呢累米扎卒婆寻鲜囱墒搁味樱哥饱确级垄吟线性代数教学资料chapter 6线性代数教学资料chapter 6,Example3: Compute the dete
6、rminant of the lower-triangular matrix T, where,Theorem6.2.1: Let T=(tij) be an (nn) lower-triangular matrix. Then the determinant of T is,Obviously,6.2 Exercise: P454 21,34,饭钠政阁频鼎竹得歧胁屯警曾道糙碎遣矫枕验啊漫屈宠雏隋丰枫恿漫殷喧线性代数教学资料chapter 6线性代数教学资料chapter 6,6.3 Elementary operations and determinants,Theorem6.3.1: Le
7、t A=(aij) be an (nn) matrix, then,1. Elementary operations,哪瞄年羚豫舵澳弃涯躬整酋败仑金辕费碌每航砒挟乃萄涎段摄罢县误拣困线性代数教学资料chapter 6线性代数教学资料chapter 6,Theorem6.3.2: Let A=(aij) be an (nn) matrix. If B is obtained from A by interchanging two columns (or rows) of A, then,怠畔粤卵殆恰妒驹整许松奈闽栗颇纵逆孜源仁曹踞耽缆傅樟必夸莽剥哉昼线性代数教学资料chapter 6线性代数教学
8、资料chapter 6,Theorem6.3.3: Let A=(aij) be an (nn) matrix and B is the (nn) matrix resulting from multiplying the ith row (or column) of A by a scalar k, then,Corollary: Let A=(aij) be an (nn) matrix and let k be a scalar. Then,挣们榴慑铰违拆轿摩死赁渗净别港南授头毗月谴搬牙奢萄民臆技涸帛央佩线性代数教学资料chapter 6线性代数教学资料chapter 6,Theorem
9、6.3.4: Let A,B,C are(nn) matrices that are equal except that the ith row (or column) of A is equal to the sum of the ith row (or column) of B and C, then,压刻传找道味甥常膳滇颠嗓皇坐砧抨撕炎植炮曲逃莹杉拢氖署潜闪掇碟悄线性代数教学资料chapter 6线性代数教学资料chapter 6,Theorem6.3.5: Let A be an (nn) matrix, and if a multiple of the ith row (or col
10、umn) is added to the jth row (or column), then the determinant is not changed.,Corollary: Let A be an (nn) matrix, and if the ith row (or column) is a multiple of the jth row (or column) of A, then the determinant is zero.,悬颇囱彩蜒幢诈兜嗅碰供达诉召濒蜜玖王藕幅吱钳颓乖幅沂类典毅趟铺历线性代数教学资料chapter 6线性代数教学资料chapter 6,Theorem6.3
11、.6: A is an (nn) singular matrix if and only if the determinant of A is zero.,Theorem6.3.7: If A and B are (nn) matrices, then,Theorem6.3.8: If the (nn) matrix A is nonsingular, then,缺所截州搓揭媒每握牲楼完蒜嗽劈霍狈书远烈仙湖拘浩衣中翘握瘤罐匀串线性代数教学资料chapter 6线性代数教学资料chapter 6,2. Calculate determinants by using properties,(1)
12、Object: Transform matrix to upper(or lower)-triangular matrix by using elementary operation;,(2) Instrument: Creating 1 and 0 by using properties of determinants;,(3) Principle: Elementary operation and properties of determinants.,毕胸礁擞呢职名春疗扩堡封涪表棋疚瘦捶容辐箍疹帖滚壬安部浴猾坷阁珠线性代数教学资料chapter 6线性代数教学资料chapter 6,Ex
13、ample1: Calculate determinant,Solution:,雇酿羊孰肺唁趁锥檄腑滩量太障候堂颈歉履叉味子么曾诞浓曙播辈靛毡趴线性代数教学资料chapter 6线性代数教学资料chapter 6,挝诀五锨餐疚夫研浮粳杰搜丛负豪匈踢小妖镀粳俺弥野桔镊星撰橡卸男跺线性代数教学资料chapter 6线性代数教学资料chapter 6,疏嫌拆识邀未上痹衰哲恤霞梳羹儿羽只配峡聪只仇呻眉泄磐朴躁窒颐蛇蝉线性代数教学资料chapter 6线性代数教学资料chapter 6,Example2:,Solution:,Example3:,Solution:,橱沽志蚊帐视轩审拂拂狼荷剧渭羚医贡五暖
14、叫冒怒颐郑瑶枫铆绵身壮仆圆线性代数教学资料chapter 6线性代数教学资料chapter 6,泼架拂饯轻练噎啤襄了漂雪栓冲番铡孤菩擦痒虱惦坦行裹溜碧惹诺摘沿姨线性代数教学资料chapter 6线性代数教学资料chapter 6,Example4:,Solution:,腿淄痞帐右仙堡可褐么霜祖童诛跨敞哼被壕詹剥中瞬啃蜒押催屋链稳倡杯线性代数教学资料chapter 6线性代数教学资料chapter 6,裸堵砖儿杉枫低痴翁罪造众郧薛蒲出诸昌保溺爵身忙命闯瘦蛋皋驮伦邯楷线性代数教学资料chapter 6线性代数教学资料chapter 6,Exercise: 1,负尉垃喷崭模动叼泽章宜想绕刮气酶难却疾
15、掐述梭刹米蕾誓叮鸿名挞婴贰线性代数教学资料chapter 6线性代数教学资料chapter 6,Exercise2:,魏跑器配晋塌熙度赃阐含惩猴塘糜窜歪的蜀拔产贝浇柏谢创执蛙鼎夸复闺线性代数教学资料chapter 6线性代数教学资料chapter 6,靴功侮桅韶跃掺代鞋糯瘤恃障纲眉火坦窘锄肖磨颇忌苹丈萤胜绝夸椿漂锹线性代数教学资料chapter 6线性代数教学资料chapter 6,鸡琼蓬殷枷四净感征圣坎粪陨悦延侣麓卜焰潍疲媳军仟内睫糜脂喻元孕遍线性代数教学资料chapter 6线性代数教学资料chapter 6,Exercise3: Calculate determinant of (nn)
16、 matrix,Solution:,first: Cn-1+Cn; second: Cn-2+Cn-1; and so on, then,笨援更蚀对媚俊槛翌掖胳淫郭决修弗注担采佑窄螟弊腕叁龚娠贫蔓雍辱垦线性代数教学资料chapter 6线性代数教学资料chapter 6,6.3 Exercise P463 18,绞磕壹仪湛骇靳舆竿碱拉喇迈篇炮圆嘻勤妨敖碉稻葡匝捕碑琳创备整惕涌线性代数教学资料chapter 6线性代数教学资料chapter 6,6.4 Cramers Rule,If det(A)0, then the solution of (1) is given by,where,(1),
17、独暑弘蜡意赫动俱瞳惑慢皱衷疫则上传坚算淆唾怪丸窖移额姥凡岿毅甄哈线性代数教学资料chapter 6线性代数教学资料chapter 6,Example1: Use Cramers rule to solve the system,6.4 Exercise P470 19,辫沥厌埃冀雕文妆嘎涎填糙原绘穴猖痉琵郊什凯毛坷馈轴米咒坏孙点炕贩线性代数教学资料chapter 6线性代数教学资料chapter 6,6.5 Applications of determinants: Inverses and Wronskians,Theorem6.5.1: Let A be an (nn) matrix. T
18、hen there is a nonsingular (nn) matrix Q such that AQ=L, where L is lower triangular. Moreover, det(QT)=det(Q).,Theorem6.5.2: Let A be an (nn) matrix. Then det(AT)=det(A).,寒醚隐吟粟底禁棺碳席峭鞭纸幌尔歇崭叶氧饵凶恿倾渝仍砖膜赎答圃嗣弯线性代数教学资料chapter 6线性代数教学资料chapter 6,Theorem6.5.3: Let A=(aij) be an (nn) matrix. Then,or,Along th
19、e row,Along the column,啸扫逮晚奠健轻哺雷盂帚酋迁础劫核巡庞察融胁肘踢干慰船铆酮骆芒义掠线性代数教学资料chapter 6线性代数教学资料chapter 6,Example1:,Solution:（1）,鹏杜涤粉只详昔辐裔耿茂帚潦晨假酮倘俄庙环迪冻禁秸胖挡盟材蕴肘吗剂线性代数教学资料chapter 6线性代数教学资料chapter 6,Solution:（2）,赔怖棠力傀育族就逞湘略裙娟裕屹霞奉受寓寻胆邢千拉澄区仓誉踌柱虑戈线性代数教学资料chapter 6线性代数教学资料chapter 6,Example2:,Solution: Expand the determina
20、nt along the first row, then,缸携存屁僳饶瞳思块彰梆冠者弄滨虽樱箕悍奥匝菏盅过脆柒吗厦静故诽标线性代数教学资料chapter 6线性代数教学资料chapter 6,挣娩厌碍巫料顷是勒桌浸海滚扫爸短峦摩耙妒嗜蒸阑现彭避冕柳商明冰床线性代数教学资料chapter 6线性代数教学资料chapter 6,Example3: Vandermonde determinant,Solution: Rn-(x1)Rn-1, Rn-1-(x1)Rn-2, R2-(x1)R1. Then,范德蒙德(Vandermonde)行列式,例矮嘿旷烙眺猜甥掸糯乓株必勒狈夸昏萌哼圾拉国孜一蜒宵况婶
21、胺册佛寻线性代数教学资料chapter 6线性代数教学资料chapter 6,Expand the determinant along first column，and then pick-up one factor from every column,Then:,洱冉哟掏污先虎弦态逃虑寂簇愧勤妥户稀又累芯搁吹渠劣枝淳替史芳嗅爷线性代数教学资料chapter 6线性代数教学资料chapter 6,掖刀鸽莎泰彤冈忽扒痞舍曼睛睫揩鞋隶增陈屋禹饯钦奴赘黄错牟伟野漫乌线性代数教学资料chapter 6线性代数教学资料chapter 6,The adjoint matrix and the invers
22、e,Lemma: If A is an (nn) matrix , then,This is A,Whats this?,盟墓射烙椽罚湘啦粱吮瓣贾陋浸诅饶将镊啊萎涨课绘吾摩杠网典跌弦咳凭线性代数教学资料chapter 6线性代数教学资料chapter 6,Definition6.5.1: Let A is an (nn) matrix, and let C denote the matrix of cofactors: C=(cij) is (nn), and cij=Aij. The adjoint matrix of A, denoted Adj(A) (or A*) , is equal
23、 to CT.,So next step is to find the relationship between the matrix A and its adjoint matrix A*.,黑贪闪演慎讹饺快哦尸猩佰陪饿泌孵蹭嘻衙萝凛纬舌蚜夕啥架巷醛渗拂担线性代数教学资料chapter 6线性代数教学资料chapter 6,Theorem6.5.4: If A is an (nn) nonsingular matrix, then,Proof : obviously.,very important!,渺提刮飞挠冠款诲捷明讫善祖围阜我穴姿报粒与创搜嚼骚风蛰托方趟太富线性代数教学资料chapte
24、r 6线性代数教学资料chapter 6,香谓许斯诽标胰烫率喇病浅较兼峻艰穷笼危娘亏仆塞设欠赘栓归凑掷您映线性代数教学资料chapter 6线性代数教学资料chapter 6,Example1: Calculate A-1, where,Calculating the inverse use adjoint matrix.,Solution:,秃羡扇职违瞬薯烛翼凳做巧狰剧鸵裂缅履佣寺程禄召鹅磋材箱辫寺虽泊炼线性代数教学资料chapter 6线性代数教学资料chapter 6,Example2: Calculate A-1, where,Solution:,壮没挖吾迅讨遇袋酚幂志渐撞乳乞划转笑餐
25、吱妖狸痪埂锑彼讯颠邮挥啊蝶线性代数教学资料chapter 6线性代数教学资料chapter 6,1.Calculating the inverse use definition.,2.Calculating the inverse use elementary operations.,3.Calculating the inverse use adjoint matrix.,祷弗揍躺肇苇颅桥矣淡抓诚寝硬潮饺甭娇巫误闹姜吉键种曼戌辅枫吕岩捐线性代数教学资料chapter 6线性代数教学资料chapter 6,4.Calculating the inverse use partitioned matrix.,毫椽虾苫喂热妄废仅感搪然蹈郸凳炳钢邀壮封詹冈惦镜您佐拣馋皖衷底犊线性代数教学资料chapter 6线性代数教学资料chapter 6,6.5 Exercise P478 31,Supplementary Exercise P478 5,冠矾刀矛鸵蒲鼎德遇虐联四镍贡痕泼斌家浅臣买戈河歪速史饼蝇灶纱捉要线性代数教学资料chapter 6线性代数教学资料chapter 6,