Formula for the inverse matrix. Cramer s rule. Review: 3 3 determinants can be computed expanding by any row or column
|
|
- Ethelbert Powell
- 6 years ago
- Views:
Transcription
1 Math 20F Linear Algebra Lecture 18 1 Determinants, n n Review: The 3 3 case Slide 1 Determinants n n (Expansions by rows and columns Relation with Gauss elimination matrices: Properties) Formula for the inverse matrix Cramer s rule (To solve nonhomogeneous systems of equations) Review: 3 3 determinants can be computed expanding by any row or column Slide 2 Claim 1 The determinant of a 3 3 matrix can be computed in terms of 2 2 determinants, expanding by any column or row, using the following sign convention for the addition, , Sign of coefficient aij is ( 1)i+j
2 Math 20F Linear Algebra Lecture 18 2 Review: Main properties of 3 3 determinants Let A = [a 1, a 2, a 3 ] be a 3 3 matrix Let c be a 3-vector Slide 3 det([a 1, a 1, a 2 ]) = 0 det([a 1, a 2, a 3 ]) = det([a 2, a 1, a 3 ]) det([a 1, a 2, a 3 ]) = det([a 1, a 3, a 2 ]) det([ca 1, a 2, a 2 ]) = c det([a 1, a 2, a 3 ]) det([a 1 + c, a 2, a 3 ]) = det([a 1, a 2, a 3 ]) + det([c, a 2, a 3 ]) Review: Important results concerning 3 3 determinants Slide 4 Theorem 1 Let A, B be 3 3 matrices Then, A is invertible det(a) 0 a 1, a 2, a 3 are ld det([a 1, a 2, a 3 ]) = 0 det(a) = det(a T ) det(ab) = det(a) det(b)
3 Math 20F Linear Algebra Lecture 18 3 Gauss elimination can be used to compute determinants! Theorem 2 Let A be a 3 3 matrix Slide 5 Let B be the result of adding to a row in A a multiple of another row in A Then, det(b) = det(a) Let B be the result of interchanging two rows in A Then, det(b) = det(a) Let B be the result of multiply a row in A by a number k Then, det(a) = (1/k) det(b) Notation needed for the n n case Slide a 11 a 1j a 1n A ij = a i1 a ij a in, a n1 a nj a nn eliminate the column j, and the row i, Sign of coefficient a ij is ( 1) i+j 7 5
4 Math 20F Linear Algebra Lecture 18 4 Determinant n n: expansion by the first row Definition 1 The determinant of an n n matrix A = [a ij ] is given by Slide 7 det(a) = det(a 11)a 11 det(a 12)a ( 1) 1+n det(a 1n)a 1n, = nx ( 1) 1+j det(a 1j) a 1j = j=1 nx a 1j C 1j, where C ij is called the cofactor of a matrix A and is the number given by C ij = ( 1) i+j det(a ij ) j=1 Determinants can be computed expanding along any row or any column Slide 8 Theorem 3 The determinant of an n n matrix A = [a ij ] can be computed by an expansion along any row or along any column That is, n det(a) = C ij a ij, for any i = 1,, n, = j=1 n C ij a ij, for any j = 1,, n i=1
5 Math 20F Linear Algebra Lecture 18 5 Use the row or column with the most number of zeros to compute the determinant Theorem 4 The determinant of a triangular matrix is the product of its diagonal elements Slide = (1) = = = (1) = = 15 Main properties of n n determinants Slide 10 Let A = [a 1,, a n ] be a n n matrix Let c be a n-vector det([a 1,, a i,, a i,, a n]) = 0 det([a 1,, a i,, a j,, a n]) = det([a 1,, a j,, a i,, a n]) det([a 1,, a j + c,, a n]) = det([a 1,, a j, a n]) + det([a 1,, c,, a n]) det([a 1,, ca j,, a n]) = c det([a 1,, a j,, a n])
6 Math 20F Linear Algebra Lecture 18 6 Important results concerning n n determinants Theorem 5 Let A, B be n n matrices Then, Slide 11 A is invertible det(a) 0 a 1,, a n are ld det([a 1,, a n ]) = 0 det(a) = det(a T ) det(ab) = det(a) det(b) The properties of the determinant on the column vectors of A and the property det(a) = det(a T ) imply the following results on the rows of A Theorem 6 (Determinants and elementary row operations) Let A be a n n matrix Let B be the result of adding to a row in A a multiple of another row in A Then, det(b) = det(a) Let B be the result of interchanging two rows in A Then, det(b) = det(a) Let B be the result of multiply a row in A by a number k Then, det(b) = k det(a) Determinant and Gauss elimination operations Theorem 7 If E represents an elementary row operation and A is an n n matrix, then det(ea) = det(e) det(a) The proof is to compute the determinant of every elementary row operation matrix, E, and then use the previous theorem Theorem 8 (Determinant of a product) If A, B are arbitrary n n matrices, then det(ab) = det(a) det(b) Determinant of a product of matrices Proof: If A is not invertible, then AB is not invertible, then the theorem holds, because 0 = det(ab) = det(a) det(b) = 0 Suppose that A is invertible Then there exist elementary row operations E k,, E 1 such that A = E k E 1
7 Math 20F Linear Algebra Lecture 18 7 Then, det(ab) = det(e k E 1 B), = det(e k ) det(e k 1 E 1 B), = det(e k ) det(e 1 ) det(b), = det(e k E 1 ) det(b), = det(a) det(b) Formula for the inverse matrix Slide 12 Theorem 9 Let A = [a ij ] be an n n matrix, C ij = ( 1) i+j det(a ij ) be the ijth cofactor, and = det(a) Then the inverse matrix A 1 is given by C 11 C 21 C n1 A 1 = 1 C 12 C 22 C n2 C 1n C 2n C nn Formula for the inverse matrix Proof: It is a straightforward computation Let us denote B the matrix with components (B) ij = C ji / Then, a 11 a 12 a 1n C 11 C 21 C n1 a 21 a 22 a 2n 1 C 12 C 22 C n2 AB = a n1 a n2 a nn C 1n C 2n C nn Compute each component of the product AB (AB) 11 = 1 (C 11a 11 + C 12 a C 1n a 1n ) = 1, because the factor in the numerator in the right hand side is precisely det(a) = The second component is given by (AB) 12 = 1 (C 11a 21 + C 12 a C 1n a 2n )
8 Math 20F Linear Algebra Lecture 18 8 The factor between brackets in the right hand side is an expansion by the first row of the determinant of a matrix whose first row is a 21, a 22, a 2n That is, (AB) 12 = 1 a 21 a 22 a 2n a 21 a 22 a 2n a 31 a 32 a 3n a n1 a n2 a nn An analogous calculation shows that (AB) ij is given by = 0 (AB) ij = 1 (C j1a i1 + C j2 a i2 + + C jn a in ), The factor between brackets in the right hand side is an expansion by the j row of the determinant of a matrix whose j row is is the i row of A, That is, a i1, a i2, a in a 11 a 12 a 1n (AB) ij = 1 a i1 a i2 a in a n1 a n2 a nn in the j-row Therefore, when i j the factor between brackets is the determinant of a matrix with two identical rows, so (AB) ij = 0 for i j If i = j, the the that factor is precisely det(a), then (AB) ii = 1 Summarizing, (AB) ij = a 11 a 12 a 1n 1 a i1 a i2 a in in the j-row a n1 a n2 a nn = I ij Repeat this calculation for BA
9 Math 20F Linear Algebra Lecture 18 9 Cramer s rule is a formula to solve nonhomogeneous linear equations Slide 13 Theorem 10 Let A be an invertible n n matrix, so the system of linear equations Ax = b has a unique solution for every vector b Then the components x i of the solution x are given by x i = 1 det(a i(b)) where we introduced the matrix A i(b) = [a 1,, b,, a n], with b placed in the i-column Proof: On the one hand, A invertible means that the solution can be written as x = A 1 b From the formula of the inverse matrix one obtains x i = 1 (C 1i b 1 + C 2i b C ni b n ), Slide 14 where b i are the components of b On the other hand, if one expands the det(a i (b)) by the i row one gets det(a i (b)) = (C 1i b 1 + C 2i b C ni b n )
Components and change of basis
Math 20F Linear Algebra Lecture 16 1 Components and change of basis Slide 1 Review: Isomorphism Review: Components in a basis Unique representation in a basis Change of basis Review: Isomorphism Definition
More informationDeterminants Chapter 3 of Lay
Determinants Chapter of Lay Dr. Doreen De Leon Math 152, Fall 201 1 Introduction to Determinants Section.1 of Lay Given a square matrix A = [a ij, the determinant of A is denoted by det A or a 11 a 1j
More informationProperties of the Determinant Function
Properties of the Determinant Function MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Overview Today s discussion will illuminate some of the properties of the determinant:
More informationChapter 2:Determinants. Section 2.1: Determinants by cofactor expansion
Chapter 2:Determinants Section 2.1: Determinants by cofactor expansion [ ] a b Recall: The 2 2 matrix is invertible if ad bc 0. The c d ([ ]) a b function f = ad bc is called the determinant and it associates
More informationEvaluating Determinants by Row Reduction
Evaluating Determinants by Row Reduction MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Objectives Reduce a matrix to row echelon form and evaluate its determinant.
More informationLemma 8: Suppose the N by N matrix A has the following block upper triangular form:
17 4 Determinants and the Inverse of a Square Matrix In this section, we are going to use our knowledge of determinants and their properties to derive an explicit formula for the inverse of a square matrix
More informationLinear Systems and Matrices
Department of Mathematics The Chinese University of Hong Kong 1 System of m linear equations in n unknowns (linear system) a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.......
More informationDeterminants by Cofactor Expansion (III)
Determinants by Cofactor Expansion (III) Comment: (Reminder) If A is an n n matrix, then the determinant of A can be computed as a cofactor expansion along the jth column det(a) = a1j C1j + a2j C2j +...
More informationMath 240 Calculus III
The Calculus III Summer 2015, Session II Wednesday, July 8, 2015 Agenda 1. of the determinant 2. determinants 3. of determinants What is the determinant? Yesterday: Ax = b has a unique solution when A
More informationLinear Algebra. Matrices Operations. Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0.
Matrices Operations Linear Algebra Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0 The rectangular array 1 2 1 4 3 4 2 6 1 3 2 1 in which the
More informationMATH 2050 Assignment 8 Fall [10] 1. Find the determinant by reducing to triangular form for the following matrices.
MATH 2050 Assignment 8 Fall 2016 [10] 1. Find the determinant by reducing to triangular form for the following matrices. 0 1 2 (a) A = 2 1 4. ANS: We perform the Gaussian Elimination on A by the following
More informationMATH 2030: EIGENVALUES AND EIGENVECTORS
MATH 2030: EIGENVALUES AND EIGENVECTORS Determinants Although we are introducing determinants in the context of matrices, the theory of determinants predates matrices by at least two hundred years Their
More informationMATH 1210 Assignment 4 Solutions 16R-T1
MATH 1210 Assignment 4 Solutions 16R-T1 Attempt all questions and show all your work. Due November 13, 2015. 1. Prove using mathematical induction that for any n 2, and collection of n m m matrices A 1,
More informationDeterminants. Samy Tindel. Purdue University. Differential equations and linear algebra - MA 262
Determinants Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra by Goode and Annin Samy T. Determinants Differential equations
More informationThe determinant. Motivation: area of parallelograms, volume of parallepipeds. Two vectors in R 2 : oriented area of a parallelogram
The determinant Motivation: area of parallelograms, volume of parallepipeds Two vectors in R 2 : oriented area of a parallelogram Consider two vectors a (1),a (2) R 2 which are linearly independent We
More informationChapter 4. Determinants
4.2 The Determinant of a Square Matrix 1 Chapter 4. Determinants 4.2 The Determinant of a Square Matrix Note. In this section we define the determinant of an n n matrix. We will do so recursively by defining
More informationdet(ka) = k n det A.
Properties of determinants Theorem. If A is n n, then for any k, det(ka) = k n det A. Multiplying one row of A by k multiplies the determinant by k. But ka has every row multiplied by k, so the determinant
More informationTOPIC III LINEAR ALGEBRA
[1] Linear Equations TOPIC III LINEAR ALGEBRA (1) Case of Two Endogenous Variables 1) Linear vs. Nonlinear Equations Linear equation: ax + by = c, where a, b and c are constants. 2 Nonlinear equation:
More informationLinear Algebra and Vector Analysis MATH 1120
Faculty of Engineering Mechanical Engineering Department Linear Algebra and Vector Analysis MATH 1120 : Instructor Dr. O. Philips Agboola Determinants and Cramer s Rule Determinants If a matrix is square
More informationMATH2210 Notebook 2 Spring 2018
MATH2210 Notebook 2 Spring 2018 prepared by Professor Jenny Baglivo c Copyright 2009 2018 by Jenny A. Baglivo. All Rights Reserved. 2 MATH2210 Notebook 2 3 2.1 Matrices and Their Operations................................
More informationLecture 10: Determinants and Cramer s Rule
Lecture 0: Determinants and Cramer s Rule The determinant and its applications. Definition The determinant of a square matrix A, denoted by det(a) or A, is a real number, which is defined as follows. -by-
More informationMAC Module 3 Determinants. Learning Objectives. Upon completing this module, you should be able to:
MAC 2 Module Determinants Learning Objectives Upon completing this module, you should be able to:. Determine the minor, cofactor, and adjoint of a matrix. 2. Evaluate the determinant of a matrix by cofactor
More informationElementary Matrices. which is obtained by multiplying the first row of I 3 by -1, and
Elementary Matrices In this special handout of material not contained in the text, we introduce the concept of elementary matrix. Elementary matrices are useful in several ways that will be shown in this
More informationENGR-1100 Introduction to Engineering Analysis. Lecture 21. Lecture outline
ENGR-1100 Introduction to Engineering Analysis Lecture 21 Lecture outline Procedure (algorithm) for finding the inverse of invertible matrix. Investigate the system of linear equation and invertibility
More informationLinear Algebra: Lecture notes from Kolman and Hill 9th edition.
Linear Algebra: Lecture notes from Kolman and Hill 9th edition Taylan Şengül March 20, 2019 Please let me know of any mistakes in these notes Contents Week 1 1 11 Systems of Linear Equations 1 12 Matrices
More informationENGR-1100 Introduction to Engineering Analysis. Lecture 21
ENGR-1100 Introduction to Engineering Analysis Lecture 21 Lecture outline Procedure (algorithm) for finding the inverse of invertible matrix. Investigate the system of linear equation and invertibility
More informationII. Determinant Functions
Supplemental Materials for EE203001 Students II Determinant Functions Chung-Chin Lu Department of Electrical Engineering National Tsing Hua University May 22, 2003 1 Three Axioms for a Determinant Function
More informationMath 320, spring 2011 before the first midterm
Math 320, spring 2011 before the first midterm Typical Exam Problems 1 Consider the linear system of equations 2x 1 + 3x 2 2x 3 + x 4 = y 1 x 1 + 3x 2 2x 3 + 2x 4 = y 2 x 1 + 2x 3 x 4 = y 3 where x 1,,
More informationMATH 323 Linear Algebra Lecture 6: Matrix algebra (continued). Determinants.
MATH 323 Linear Algebra Lecture 6: Matrix algebra (continued). Determinants. Elementary matrices Theorem 1 Any elementary row operation σ on matrices with n rows can be simulated as left multiplication
More informationMAT 1332: CALCULUS FOR LIFE SCIENCES. Contents. 1. Review: Linear Algebra II Vectors and matrices Definition. 1.2.
MAT 1332: CALCULUS FOR LIFE SCIENCES JING LI Contents 1 Review: Linear Algebra II Vectors and matrices 1 11 Definition 1 12 Operations 1 2 Linear Algebra III Inverses and Determinants 1 21 Inverse Matrices
More informationEXAM. Exam #1. Math 2360, Second Summer Session, April 24, 2001 ANSWERS
i i EXAM Exam #1 Math 2360, Second Summer Session, 2002 April 24, 2001 ANSWERS i 50 pts. Problem 1. In each part you are given the augmented matrix of a system of linear equations, with the coefficent
More informationUndergraduate Mathematical Economics Lecture 1
Undergraduate Mathematical Economics Lecture 1 Yu Ren WISE, Xiamen University September 15, 2014 Outline 1 Courses Description and Requirement 2 Course Outline ematical techniques used in economics courses
More informationGraduate Mathematical Economics Lecture 1
Graduate Mathematical Economics Lecture 1 Yu Ren WISE, Xiamen University September 23, 2012 Outline 1 2 Course Outline ematical techniques used in graduate level economics courses Mathematics for Economists
More informationNotes on Determinants and Matrix Inverse
Notes on Determinants and Matrix Inverse University of British Columbia, Vancouver Yue-Xian Li March 17, 2015 1 1 Definition of determinant Determinant is a scalar that measures the magnitude or size of
More informationLectures on Linear Algebra for IT
Lectures on Linear Algebra for IT by Mgr. Tereza Kovářová, Ph.D. following content of lectures by Ing. Petr Beremlijski, Ph.D. Department of Applied Mathematics, VSB - TU Ostrava Czech Republic 11. Determinants
More informationMethods for Solving Linear Systems Part 2
Methods for Solving Linear Systems Part 2 We have studied the properties of matrices and found out that there are more ways that we can solve Linear Systems. In Section 7.3, we learned that we can use
More informationMaterials engineering Collage \\ Ceramic & construction materials department Numerical Analysis \\Third stage by \\ Dalya Hekmat
Materials engineering Collage \\ Ceramic & construction materials department Numerical Analysis \\Third stage by \\ Dalya Hekmat Linear Algebra Lecture 2 1.3.7 Matrix Matrix multiplication using Falk s
More informationLECTURE 4: DETERMINANT (CHAPTER 2 IN THE BOOK)
LECTURE 4: DETERMINANT (CHAPTER 2 IN THE BOOK) Everything with is not required by the course syllabus. Idea Idea: for each n n matrix A we will assign a real number called det(a). Properties: det(a) 0
More informationA = , A 32 = n ( 1) i +j a i j det(a i j). (1) j=1
Lecture Notes: Determinant of a Square Matrix Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk 1 Determinant Definition Let A [a ij ] be an
More informationDeterminants. Beifang Chen
Determinants Beifang Chen 1 Motivation Determinant is a function that each square real matrix A is assigned a real number, denoted det A, satisfying certain properties If A is a 3 3 matrix, writing A [u,
More informationElementary maths for GMT
Elementary maths for GMT Linear Algebra Part 2: Matrices, Elimination and Determinant m n matrices The system of m linear equations in n variables x 1, x 2,, x n a 11 x 1 + a 12 x 2 + + a 1n x n = b 1
More informationEquality: Two matrices A and B are equal, i.e., A = B if A and B have the same order and the entries of A and B are the same.
Introduction Matrix Operations Matrix: An m n matrix A is an m-by-n array of scalars from a field (for example real numbers) of the form a a a n a a a n A a m a m a mn The order (or size) of A is m n (read
More informationNOTES FOR LINEAR ALGEBRA 133
NOTES FOR LINEAR ALGEBRA 33 William J Anderson McGill University These are not official notes for Math 33 identical to the notes projected in class They are intended for Anderson s section 4, and are 2
More informationMath 313 (Linear Algebra) Exam 2 - Practice Exam
Name: Student ID: Section: Instructor: Math 313 (Linear Algebra) Exam 2 - Practice Exam Instructions: For questions which require a written answer, show all your work. Full credit will be given only if
More informationDeterminants. Recall that the 2 2 matrix a b c d. is invertible if
Determinants Recall that the 2 2 matrix a b c d is invertible if and only if the quantity ad bc is nonzero. Since this quantity helps to determine the invertibility of the matrix, we call it the determinant.
More informationOR MSc Maths Revision Course
OR MSc Maths Revision Course Tom Byrne School of Mathematics University of Edinburgh t.m.byrne@sms.ed.ac.uk 15 September 2017 General Information Today JCMB Lecture Theatre A, 09:30-12:30 Mathematics revision
More informationMATH 300, Second Exam REVIEW SOLUTIONS. NOTE: You may use a calculator for this exam- You only need something that will perform basic arithmetic.
MATH 300, Second Exam REVIEW SOLUTIONS NOTE: You may use a calculator for this exam- You only need something that will perform basic arithmetic. [ ] [ ] 2 2. Let u = and v =, Let S be the parallelegram
More information5.3 Determinants and Cramer s Rule
304 53 Determinants and Cramer s Rule Unique Solution of a 2 2 System The 2 2 system (1) ax + by = e, cx + dy = f, has a unique solution provided = ad bc is nonzero, in which case the solution is given
More informationMATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam. Topics
MATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam This study sheet will not be allowed during the test Books and notes will not be allowed during the test Calculators and cell phones
More informationChapter 4 - MATRIX ALGEBRA. ... a 2j... a 2n. a i1 a i2... a ij... a in
Chapter 4 - MATRIX ALGEBRA 4.1. Matrix Operations A a 11 a 12... a 1j... a 1n a 21. a 22.... a 2j... a 2n. a i1 a i2... a ij... a in... a m1 a m2... a mj... a mn The entry in the ith row and the jth column
More informationReview for Exam Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions.
Review for Exam. Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions. x + y z = 2 x + 2y + z = 3 x + y + (a 2 5)z = a 2 The augmented matrix for
More information7.6 The Inverse of a Square Matrix
7.6 The Inverse of a Square Matrix Copyright Cengage Learning. All rights reserved. What You Should Learn Verify that two matrices are inverses of each other. Use Gauss-Jordan elimination to find inverses
More informationSPRING OF 2008 D. DETERMINANTS
18024 SPRING OF 2008 D DETERMINANTS In many applications of linear algebra to calculus and geometry, the concept of a determinant plays an important role This chapter studies the basic properties of determinants
More informationMatrices and Linear Algebra
Contents Quantitative methods for Economics and Business University of Ferrara Academic year 2017-2018 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2
More informationMATH Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product.
MATH 311-504 Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product. Determinant is a scalar assigned to each square matrix. Notation. The determinant of a matrix A = (a ij
More informationc c c c c c c c c c a 3x3 matrix C= has a determinant determined by
Linear Algebra Determinants and Eigenvalues Introduction: Many important geometric and algebraic properties of square matrices are associated with a single real number revealed by what s known as the determinant.
More informationChapter 2. Square matrices
Chapter 2. Square matrices Lecture notes for MA1111 P. Karageorgis pete@maths.tcd.ie 1/18 Invertible matrices Definition 2.1 Invertible matrices An n n matrix A is said to be invertible, if there is a
More informationIntroduction to Matrices and Linear Systems Ch. 3
Introduction to Matrices and Linear Systems Ch. 3 Doreen De Leon Department of Mathematics, California State University, Fresno June, 5 Basic Matrix Concepts and Operations Section 3.4. Basic Matrix Concepts
More information[ Here 21 is the dot product of (3, 1, 2, 5) with (2, 3, 1, 2), and 31 is the dot product of
. Matrices A matrix is any rectangular array of numbers. For example 3 5 6 4 8 3 3 is 3 4 matrix, i.e. a rectangular array of numbers with three rows four columns. We usually use capital letters for matrices,
More informationLesson 3. Inverse of Matrices by Determinants and Gauss-Jordan Method
Module 1: Matrices and Linear Algebra Lesson 3 Inverse of Matrices by Determinants and Gauss-Jordan Method 3.1 Introduction In lecture 1 we have seen addition and multiplication of matrices. Here we shall
More informationMATH 240 Spring, Chapter 1: Linear Equations and Matrices
MATH 240 Spring, 2006 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.1 1.6, 2.1 2.2, 3.2 3.8, 4.3 4.5, 5.1 5.3, 5.5, 6.1 6.5, 7.1 7.2, 7.4 DEFINITIONS Chapter 1: Linear
More informationIntroduction. Vectors and Matrices. Vectors [1] Vectors [2]
Introduction Vectors and Matrices Dr. TGI Fernando 1 2 Data is frequently arranged in arrays, that is, sets whose elements are indexed by one or more subscripts. Vector - one dimensional array Matrix -
More informationLinear Algebraic Equations
Linear Algebraic Equations 1 Fundamentals Consider the set of linear algebraic equations n a ij x i b i represented by Ax b j with [A b ] [A b] and (1a) r(a) rank of A (1b) Then Axb has a solution iff
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS AND MATRICES Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationMATH 106 LINEAR ALGEBRA LECTURE NOTES
MATH 6 LINEAR ALGEBRA LECTURE NOTES FALL - These Lecture Notes are not in a final form being still subject of improvement Contents Systems of linear equations and matrices 5 Introduction to systems of
More informationLINEAR ALGEBRA WITH APPLICATIONS
SEVENTH EDITION LINEAR ALGEBRA WITH APPLICATIONS Instructor s Solutions Manual Steven J. Leon PREFACE This solutions manual is designed to accompany the seventh edition of Linear Algebra with Applications
More informationMath Computation Test 1 September 26 th, 2016 Debate: Computation vs. Theory Whatever wins, it ll be Huuuge!
Math 5- Computation Test September 6 th, 6 Debate: Computation vs. Theory Whatever wins, it ll be Huuuge! Name: Answer Key: Making Math Great Again Be sure to show your work!. (8 points) Consider the following
More informationChapter 3. Determinants and Eigenvalues
Chapter 3. Determinants and Eigenvalues 3.1. Determinants With each square matrix we can associate a real number called the determinant of the matrix. Determinants have important applications to the theory
More informationLinear Algebra Section 2.6 : LU Decomposition Section 2.7 : Permutations and transposes Wednesday, February 13th Math 301 Week #4
Linear Algebra Section. : LU Decomposition Section. : Permutations and transposes Wednesday, February 1th Math 01 Week # 1 The LU Decomposition We learned last time that we can factor a invertible matrix
More informationMatrix Algebra Determinant, Inverse matrix. Matrices. A. Fabretti. Mathematics 2 A.Y. 2015/2016. A. Fabretti Matrices
Matrices A. Fabretti Mathematics 2 A.Y. 2015/2016 Table of contents Matrix Algebra Determinant Inverse Matrix Introduction A matrix is a rectangular array of numbers. The size of a matrix is indicated
More information4. Determinants.
4. Determinants 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 2 2 determinant 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 3 3 determinant 4.1.
More informationMTH 464: Computational Linear Algebra
MTH 464: Computational Linear Algebra Lecture Outlines Exam 2 Material Prof. M. Beauregard Department of Mathematics & Statistics Stephen F. Austin State University March 2, 2018 Linear Algebra (MTH 464)
More informationIntroduction to Determinants
Introduction to Determinants For any square matrix of order 2, we have found a necessary and sufficient condition for invertibility. Indeed, consider the matrix The matrix A is invertible if and only if.
More informationSystem of Linear Equations
Math 20F Linear Algebra Lecture 2 1 System of Linear Equations Slide 1 Definition 1 Fix a set of numbers a ij, b i, where i = 1,, m and j = 1,, n A system of m linear equations in n variables x j, is given
More informationMath 4377/6308 Advanced Linear Algebra
3.1 Elementary Matrix Math 4377/6308 Advanced Linear Algebra 3.1 Elementary Matrix Operations and Elementary Matrix Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/
More information7.4. The Inverse of a Matrix. Introduction. Prerequisites. Learning Outcomes
The Inverse of a Matrix 7.4 Introduction In number arithmetic every number a 0has a reciprocal b written as a or such that a ba = ab =. Similarly a square matrix A may have an inverse B = A where AB =
More informationThis MUST hold matrix multiplication satisfies the distributive property.
The columns of AB are combinations of the columns of A. The reason is that each column of AB equals A times the corresponding column of B. But that is a linear combination of the columns of A with coefficients
More informationLecture Summaries for Linear Algebra M51A
These lecture summaries may also be viewed online by clicking the L icon at the top right of any lecture screen. Lecture Summaries for Linear Algebra M51A refers to the section in the textbook. Lecture
More information7.3. Determinants. Introduction. Prerequisites. Learning Outcomes
Determinants 7.3 Introduction Among other uses, determinants allow us to determine whether a system of linear equations has a unique solution or not. The evaluation of a determinant is a key skill in engineering
More informationMath Linear Algebra Final Exam Review Sheet
Math 15-1 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a component-wise operation. Two vectors v and w may be added together as long as they contain the same number n of
More informationDeterminants. 2.1 Determinants by Cofactor Expansion. Recall from Theorem that the 2 2 matrix
CHAPTER 2 Determinants CHAPTER CONTENTS 21 Determinants by Cofactor Expansion 105 22 Evaluating Determinants by Row Reduction 113 23 Properties of Determinants; Cramer s Rule 118 INTRODUCTION In this chapter
More informationMath Bootcamp An p-dimensional vector is p numbers put together. Written as. x 1 x =. x p
Math Bootcamp 2012 1 Review of matrix algebra 1.1 Vectors and rules of operations An p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the
More informationMath 304 Fall 2018 Exam 3 Solutions 1. (18 Points, 3 Pts each part) Let A, B, C, D be square matrices of the same size such that
Math 304 Fall 2018 Exam 3 Solutions 1. (18 Points, 3 Pts each part) Let A, B, C, D be square matrices of the same size such that det(a) = 2, det(b) = 2, det(c) = 1, det(d) = 4. 2 (a) Compute det(ad)+det((b
More informationHere are some additional properties of the determinant function.
List of properties Here are some additional properties of the determinant function. Prop Throughout let A, B M nn. 1 If A = (a ij ) is upper triangular then det(a) = a 11 a 22... a nn. 2 If a row or column
More informationANSWERS. E k E 2 E 1 A = B
MATH 7- Final Exam Spring ANSWERS Essay Questions points Define an Elementary Matrix Display the fundamental matrix multiply equation which summarizes a sequence of swap, combination and multiply operations,
More information1111: Linear Algebra I
1111: Linear Algebra I Dr. Vladimir Dotsenko (Vlad) Michaelmas Term 2015 Dr. Vladimir Dotsenko (Vlad) 1111: Linear Algebra I Michaelmas Term 2015 1 / 10 Row expansion of the determinant Our next goal is
More informationLinear Algebra (part 1) : Matrices and Systems of Linear Equations (by Evan Dummit, 2016, v. 2.02)
Linear Algebra (part ) : Matrices and Systems of Linear Equations (by Evan Dummit, 206, v 202) Contents 2 Matrices and Systems of Linear Equations 2 Systems of Linear Equations 2 Elimination, Matrix Formulation
More informationDeterminants: Uniqueness and more
Math 5327 Spring 2018 Determinants: Uniqueness and more Uniqueness The main theorem we are after: Theorem 1 The determinant of and n n matrix A is the unique n-linear, alternating function from F n n to
More informationANALYTICAL MATHEMATICS FOR APPLICATIONS 2018 LECTURE NOTES 3
ANALYTICAL MATHEMATICS FOR APPLICATIONS 2018 LECTURE NOTES 3 ISSUED 24 FEBRUARY 2018 1 Gaussian elimination Let A be an (m n)-matrix Consider the following row operations on A (1) Swap the positions any
More informationThe Determinant: a Means to Calculate Volume
The Determinant: a Means to Calculate Volume Bo Peng August 16, 2007 Abstract This paper gives a definition of the determinant and lists many of its well-known properties Volumes of parallelepipeds are
More informationLecture 8: Determinants I
8-1 MATH 1B03/1ZC3 Winter 2019 Lecture 8: Determinants I Instructor: Dr Rushworth January 29th Determinants via cofactor expansion (from Chapter 2.1 of Anton-Rorres) Matrices encode information. Often
More informationMath 317, Tathagata Basak, Some notes on determinant 1 Row operations in terms of matrix multiplication 11 Let I n denote the n n identity matrix Let E ij denote the n n matrix whose (i, j)-th entry is
More information(b) If a multiple of one row of A is added to another row to produce B then det(b) =det(a).
.(5pts) Let B = 5 5. Compute det(b). (a) (b) (c) 6 (d) (e) 6.(5pts) Determine which statement is not always true for n n matrices A and B. (a) If two rows of A are interchanged to produce B, then det(b)
More information1 Determinants. 1.1 Determinant
1 Determinants [SB], Chapter 9, p.188-196. [SB], Chapter 26, p.719-739. Bellow w ll study the central question: which additional conditions must satisfy a quadratic matrix A to be invertible, that is to
More information9 Appendix. Determinants and Cramer s formula
LINEAR ALGEBRA: THEORY Version: August 12, 2000 133 9 Appendix Determinants and Cramer s formula Here we the definition of the determinant in the general case and summarize some features Then we show how
More informationMatrix Algebra. Matrix Algebra. Chapter 8 - S&B
Chapter 8 - S&B Algebraic operations Matrix: The size of a matrix is indicated by the number of its rows and the number of its columns. A matrix with k rows and n columns is called a k n matrix. The number
More informationELEMENTARY LINEAR ALGEBRA WITH APPLICATIONS. 1. Linear Equations and Matrices
ELEMENTARY LINEAR ALGEBRA WITH APPLICATIONS KOLMAN & HILL NOTES BY OTTO MUTZBAUER 11 Systems of Linear Equations 1 Linear Equations and Matrices Numbers in our context are either real numbers or complex
More informationMTH 102A - Linear Algebra II Semester
MTH 0A - Linear Algebra - 05-6-II Semester Arbind Kumar Lal P Field A field F is a set from which we choose our coefficients and scalars Expected properties are ) a+b and a b should be defined in it )
More information2. Every linear system with the same number of equations as unknowns has a unique solution.
1. For matrices A, B, C, A + B = A + C if and only if A = B. 2. Every linear system with the same number of equations as unknowns has a unique solution. 3. Every linear system with the same number of equations
More informationMatrices Gaussian elimination Determinants. Graphics 2009/2010, period 1. Lecture 4: matrices
Graphics 2009/2010, period 1 Lecture 4 Matrices m n matrices Matrices Definitions Diagonal, Identity, and zero matrices Addition Multiplication Transpose and inverse The system of m linear equations in
More information