线性代数教学资料chapter6.ppt
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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 determin
2、ant 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 determinantsElementary operations and determinantsCramers ruleApplications 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chapter 6,6
3、.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 determinant of A is given
4、 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chapter 6线性代数教学资料chapter 6,
5、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 determinant of the lower-triangular
6、 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:Let A=(aij)be an(nn)matrix,then,1.Element
7、ary 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线性代数教学资料chapter 6,Theorem6.3.3:Let A=(aij)be an(nn)matrix
8、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,Theorem6.3.4:Let A,B,C are(nn)matrices that are equal except that the ith
9、 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 column)is added to the jth row(or column),then the determinant is not changed.,Co
10、rollary: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,Theorem6.3.6:A is an(nn)singular matrix if and only if the determinant of A is zero.,Theorem6.3.7:If
11、 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)Object:Transform matrix to upper(or lower)-triangular matrix by using elementary operation;,(2)Instrume
12、nt: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,Example1: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



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