Math 240 Calculus III


 Merilyn Lorena Fleming
 1 years ago
 Views:
Transcription
1 The Calculus III Summer 2015, Session II Wednesday, July 8, 2015
2 Agenda 1. of the determinant 2. determinants 3. of determinants
3 What is the determinant? Yesterday: Ax = b has a unique solution when A is square and nonsingular. Today: how to determine whether A is invertible. Example [ ] a b Recall that a 2 2 matrix is invertible as long as c d ad bc 0. The quantity ad bc is the determinant of this matrix and the matrix is invertible exactly when its determinant is nonzero.
4 What should the determinant be? We want to associate a number with a matrix that is zero if and only if the matrix is singular. An n n matrix is nonsingular if and only if its rank is n. For upper triangular matrices, the rank is the number of nonzero entries on the diagonal. To determine if every number in a set is nonzero, we can multiply them. The determinant of an upper triangular matrix, A = [a ij ], is the product of the elements a ii along its main diagonal. We write a 11 a 1n det(a) =.... = a 11 a 22 a nn. 0 a nn
5 What should the determinant be? What about matrices that are not upper triangular? We can make any matrix upper triangular via row reduction. So how do elementary row operations affect the determinant? M i (k) multiplies the determinant by k. (Remember that k cannot be zero.) A ij (k) does not change the determinant. P ij multiplies the determinant by 1. Let s extend these properties to all matrices. The determinant of a square matrix, A, is the determinant of any upper triangular matrix obtained from A by row reduction times 1 k for every M i(k) operation used while reducing as well as 1 for each P ij operation used.
6 determinants Example Compute det(a), where A = We need to put A in upper triangular form P A ( 2) M ( 5) A 23 ( 2) So the determinant is = ( 1)(5) = 15.
7 determinants Important Example [ ] a b Given a general 2 2 matrix, A =, compute det(a). c d [ ] [ ] a b A12( a) c a b c d 0 d bc a so a b c d = a b 0 d bc = ad bc. a This explains [ ] 1 a b = c d 1 ad bc [ ] d b when ad bc 0. c a
8 Other methods of computing determinants Theorem (Cofactor expansion) Suppose A = [a ij ] is an n n matrix. For any fixed k between 1 and n, n n det(a) = ( 1) k+j a kj det(a kj ) = ( 1) i+k a ik det(a ik ) j=1 i=1 where A ij is the (n 1) (n 1) submatrix obtained by removing the i th row and j th column from A. Example i j k a b c d e f = b e c f i a d c f j + a d b e k.
9 Other methods of computing determinants Corollary If A = [a ij ] is an n n matrix and the element a ij is the only nonzero entry in its row or column then det(a) = ( 1) i+j a ij det(a ij ). Example = = 27.
10 The Other methods of computing determinants Some of you may have learned the method of computing a 3 3 determinant by multiplying diagonals. a 11 a 12 a 13 a 11 a 12 a 21 a 22 a 23 a 21 a 22 a 31 a 32 a 33 a 31 a Be aware that this method does not work for matrices larger than 3 3.
11 of determinants Theorem (Main theorem) Suppose A is an n n matrix. The following are equivalent: A is invertible, det(a) 0. Further properties det ( A T ) = det(a). The determinant of a lower triangular matrix is also the product of the elements on the main diagonal. If A has a row or column of zeros then det(a) = 0. If two rows or columns of A are the same then det(a) = 0. det(ab) = det(a) det(b), det ( A 1) = det(a) 1. det(sa) = s n det(a). It is not true that det(a + B) = det(a) + det(b).
12 Geometric interpretation Let A be an n n matrix and a 1,..., a n be the rows or columns of A. Theorem The volume (or area, if n = 2) of the paralellepiped determined by the vectors a 1,..., a n is det(a). Source: en.wikibooks.org/wiki/linear Algebra Corollary The vectors a 1,..., a n lie in the same hyperplane if and only if det(a) = 0.
Matrix Operations: Determinant
Matrix Operations: Determinant Determinants Determinants are only applicable for square matrices. Determinant of the square matrix A is denoted as: det(a) or A Recall that the absolute value of the determinant
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 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 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 informationFormula for the inverse matrix. Cramer s rule. Review: 3 3 determinants can be computed expanding by any row or column
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
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 informationENGR1100 Introduction to Engineering Analysis. Lecture 21
ENGR1100 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 informationMATH 1210 Assignment 4 Solutions 16RT1
MATH 1210 Assignment 4 Solutions 16RT1 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 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 information1 Determinants. 1.1 Determinant
1 Determinants [SB], Chapter 9, p.188196. [SB], Chapter 26, p.719739. Bellow w ll study the central question: which additional conditions must satisfy a quadratic matrix A to be invertible, that is to
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 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 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 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 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 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 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 informationMath Linear Algebra Final Exam Review Sheet
Math 151 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a componentwise operation. Two vectors v and w may be added together as long as they contain the same number n of
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 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 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 informationAnnouncements Wednesday, October 25
Announcements Wednesday, October 25 The midterm will be returned in recitation on Friday. The grade breakdown is posted on Piazza. You can pick it up from me in office hours before then. Keep tabs on your
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 information1 Multiply Eq. E i by λ 0: (λe i ) (E i ) 2 Multiply Eq. E j by λ and add to Eq. E i : (E i + λe j ) (E i )
Direct Methods for Linear Systems Chapter Direct Methods for Solving Linear Systems PerOlof Persson persson@berkeleyedu Department of Mathematics University of California, Berkeley Math 18A Numerical
More informationTopic 15 Notes Jeremy Orloff
Topic 5 Notes Jeremy Orloff 5 Transpose, Inverse, Determinant 5. Goals. Know the definition and be able to compute the inverse of any square matrix using row operations. 2. Know the properties of inverses.
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 informationChapters 5 & 6: Theory Review: Solutions Math 308 F Spring 2015
Chapters 5 & 6: Theory Review: Solutions Math 308 F Spring 205. If A is a 3 3 triangular matrix, explain why det(a) is equal to the product of entries on the diagonal. If A is a lower triangular or diagonal
More informationMAC Module 2 Systems of Linear Equations and Matrices II. Learning Objectives. Upon completing this module, you should be able to :
MAC 0 Module Systems of Linear Equations and Matrices II Learning Objectives Upon completing this module, you should be able to :. Find the inverse of a square matrix.. Determine whether a matrix is invertible..
More informationCHAPTER 6. Direct Methods for Solving Linear Systems
CHAPTER 6 Direct Methods for Solving Linear Systems. Introduction A direct method for approximating the solution of a system of n linear equations in n unknowns is one that gives the exact solution to
More informationA matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and
Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.
More informationMath 416, Spring 2010 The algebra of determinants March 16, 2010 THE ALGEBRA OF DETERMINANTS. 1. Determinants
THE ALGEBRA OF DETERMINANTS 1. Determinants We have already defined the determinant of a 2 2 matrix: det = ad bc. We ve also seen that it s handy for determining when a matrix is invertible, and when it
More informationand let s calculate the image of some vectors under the transformation T.
Chapter 5 Eigenvalues and Eigenvectors 5. Eigenvalues and Eigenvectors Let T : R n R n be a linear transformation. Then T can be represented by a matrix (the standard matrix), and we can write T ( v) =
More informationDeterminants and Scalar Multiplication
Properties of Determinants In the last section, we saw how determinants interact with the elementary row operations. There are other operations on matrices, though, such as scalar multiplication, matrix
More informationMatrices. Math 240 Calculus III. Wednesday, July 10, Summer 2013, Session II. Matrices. Math 240. Definitions and Notation.
function Matrices Calculus III Summer 2013, Session II Wednesday, July 10, 2013 Agenda function 1. 2. function function Definition An m n matrix is a rectangular array of numbers arranged in m horizontal
More informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More informationk=1 ( 1)k+j M kj detm kj. detm = ad bc. = 1 ( ) 2 ( )+3 ( ) = = 0
4 Determinants The determinant of a square matrix is a scalar (i.e. an element of the field from which the matrix entries are drawn which can be associated to it, and which contains a surprisingly large
More informationMath Lecture 26 : The Properties of Determinants
Math 2270  Lecture 26 : The Properties of Determinants Dylan Zwick Fall 202 The lecture covers section 5. from the textbook. The determinant of a square matrix is a number that tells you quite a bit about
More informationLinear Algebra Primer
Linear Algebra Primer David Doria daviddoria@gmail.com Wednesday 3 rd December, 2008 Contents Why is it called Linear Algebra? 4 2 What is a Matrix? 4 2. Input and Output.....................................
More informationThe MatrixTree Theorem
The MatrixTree Theorem Christopher Eur March 22, 2015 Abstract: We give a brief introduction to graph theory in light of linear algebra. Our results culminates in the proof of MatrixTree Theorem. 1 Preliminaries
More informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More informationMath Camp Notes: Linear Algebra I
Math Camp Notes: Linear Algebra I Basic Matrix Operations and Properties Consider two n m matrices: a a m A = a n a nm Then the basic matrix operations are as follows: a + b a m + b m A + B = a n + b n
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 informationCalculating determinants for larger matrices
Day 26 Calculating determinants for larger matrices We now proceed to define det A for n n matrices A As before, we are looking for a function of A that satisfies the product formula det(ab) = det A det
More informationDeterminants: Elementary Row/Column Operations
Determinants: Elementary Row/Column Operations Linear Algebra Josh Engwer TTU 23 September 2015 Josh Engwer (TTU) Determinants: Elementary Row/Column Operations 23 September 2015 1 / 16 Elementary Row
More informationMath 110 Linear Algebra Midterm 2 Review October 28, 2017
Math 11 Linear Algebra Midterm Review October 8, 17 Material Material covered on the midterm includes: All lectures from Thursday, Sept. 1st to Tuesday, Oct. 4th Homeworks 9 to 17 Quizzes 5 to 9 Sections
More informationMTH 102A  Linear Algebra II Semester
MTH 0A  Linear Algebra  056II 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 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 informationUNIT 3 MATRICES  II
Algebra  I UNIT 3 MATRICES  II Structure 3.0 Introduction 3.1 Objectives 3.2 Elementary Row Operations 3.3 Rank of a Matrix 3.4 Inverse of a Matrix using Elementary Row Operations 3.5 Answers to Check
More informationLinear Algebra Practice Problems
Linear Algebra Practice Problems Math 24 Calculus III Summer 25, Session II. Determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless otherwise stated,
More informationMATH 369 Linear Algebra
Assignment # Problem # A father and his two sons are together 00 years old. The father is twice as old as his older son and 30 years older than his younger son. How old is each person? Problem # 2 Determine
More information(a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? Solution: dim N(A) 1, since rank(a) 3. Ax =
. (5 points) (a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? dim N(A), since rank(a) 3. (b) If we also know that Ax = has no solution, what do we know about the rank of A? C(A)
More informationI = i 0,
Special Types of Matrices Certain matrices, such as the identity matrix 0 0 0 0 0 0 I = 0 0 0, 0 0 0 have a special shape, which endows the matrix with helpful properties The identity matrix is an example
More informationDeterminants of 2 2 Matrices
Determinants In section 4, we discussed inverses of matrices, and in particular asked an important question: How can we tell whether or not a particular square matrix A has an inverse? We will be able
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 wellknown properties Volumes of parallelepipeds are
More informationNotes on Determinants and Matrix Inverse
Notes on Determinants and Matrix Inverse University of British Columbia, Vancouver YueXian Li March 17, 2015 1 1 Definition of determinant Determinant is a scalar that measures the magnitude or size of
More informationMath Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88
Math Camp 2010 Lecture 4: Linear Algebra Xiao Yu Wang MIT Aug 2010 Xiao Yu Wang (MIT) Math Camp 2010 08/10 1 / 88 Linear Algebra Game Plan Vector Spaces Linear Transformations and Matrices Determinant
More information= 1 and 2 1. T =, and so det A b d
Chapter 8 Determinants The founder of the theory of determinants is usually taken to be Gottfried Wilhelm Leibniz (1646 1716, who also shares the credit for inventing calculus with Sir Isaac Newton (1643
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 informationa11 a A = : a 21 a 22
Matrices The study of linear systems is facilitated by introducing matrices. Matrix theory provides a convenient language and notation to express many of the ideas concisely, and complicated formulas are
More informationFoundations of Cryptography
Foundations of Cryptography Ville Junnila, Arto Lepistö viljun@utu.fi, alepisto@utu.fi Department of Mathematics and Statistics University of Turku 2017 Ville Junnila, Arto Lepistö viljun@utu.fi, alepisto@utu.fi
More informationPhys 201. Matrices and Determinants
Phys 201 Matrices and Determinants 1 1.1 Matrices 1.2 Operations of matrices 1.3 Types of matrices 1.4 Properties of matrices 1.5 Determinants 1.6 Inverse of a 3 3 matrix 2 1.1 Matrices A 2 3 7 =! " 1
More informationLinear Algebra: Linear Systems and Matrices  Quadratic Forms and Deniteness  Eigenvalues and Markov Chains
Linear Algebra: Linear Systems and Matrices  Quadratic Forms and Deniteness  Eigenvalues and Markov Chains Joshua Wilde, revised by Isabel Tecu, Takeshi Suzuki and María José Boccardi August 3, 3 Systems
More informationA = 3 B = A 1 1 matrix is the same as a number or scalar, 3 = [3].
Appendix : A Very Brief Linear ALgebra Review Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics Very often in this course we study the shapes
More informationChapter 2: Matrices and Linear Systems
Chapter 2: Matrices and Linear Systems Paul Pearson Outline Matrices Linear systems Row operations Inverses Determinants Matrices Definition An m n matrix A = (a ij ) is a rectangular array of real numbers
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 informationHonors Advanced Mathematics Determinants page 1
Determinants page 1 Determinants For every square matrix A, there is a number called the determinant of the matrix, denoted as det(a) or A. Sometimes the bars are written just around the numbers of the
More informationDeterminants and Scalar Multiplication
Invertibility and Properties of Determinants In a previous section, we saw that the trace function, which calculates the sum of the diagonal entries of a square matrix, interacts nicely with the operations
More informationLinear Algebra. James Je Heon Kim
Linear lgebra James Je Heon Kim (jjk9columbia.edu) If you are unfamiliar with linear or matrix algebra, you will nd that it is very di erent from basic algebra or calculus. For the duration of this session,
More informationTHE ADJOINT OF A MATRIX The transpose of this matrix is called the adjoint of A That is, C C n1 C 22.. adj A. C n C nn.
8 Chapter Determinants.4 Applications of Determinants Find the adjoint of a matrix use it to find the inverse of the matrix. Use Cramer s Rule to solve a sstem of n linear equations in n variables. Use
More informationDirect Methods for Solving Linear Systems. Simon Fraser University Surrey Campus MACM 316 Spring 2005 Instructor: Ha Le
Direct Methods for Solving Linear Systems Simon Fraser University Surrey Campus MACM 316 Spring 2005 Instructor: Ha Le 1 Overview General Linear Systems Gaussian Elimination Triangular Systems The LU Factorization
More informationCalculation in the special cases n = 2 and n = 3:
9. The determinant The determinant is a function (with real numbers as values) which is defined for quadratic matrices. It allows to make conclusions about the rank and appears in diverse theorems and
More informationElementary Linear Algebra Review for Exam 3 Exam is Friday, December 11th from 1:153:15
Elementary Linear Algebra Review for Exam 3 Exam is Friday, December th from :53:5 The exam will cover sections: 6., 6.2, 7. 7.4, and the class notes on dynamical systems. You absolutely must be able
More informationDeterminants An Introduction
Determinants An Introduction Professor Je rey Stuart Department of Mathematics Paci c Lutheran University Tacoma, WA 9844 USA je rey.stuart@plu.edu The determinant is a useful function that takes a square
More informationProblem Set # 1 Solution, 18.06
Problem Set # 1 Solution, 1.06 For grading: Each problem worths 10 points, and there is points of extra credit in problem. The total maximum is 100. 1. (10pts) In Lecture 1, Prof. Strang drew the cone
More informationLecture 2: Eigenvalues and their Uses
Spectral Graph Theory Instructor: Padraic Bartlett Lecture 2: Eigenvalues and their Uses Week 3 Mathcamp 2011 As you probably noticed on yesterday s HW, we, um, don t really have any good tools for finding
More informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
More informationMath 18, Linear Algebra, Lecture C00, Spring 2017 Review and Practice Problems for Final Exam
Math 8, Linear Algebra, Lecture C, Spring 7 Review and Practice Problems for Final Exam. The augmentedmatrix of a linear system has been transformed by row operations into 5 4 8. Determine if the system
More informationChapter 3: Theory Review: Solutions Math 308 F Spring 2015
Chapter : Theory Review: Solutions Math 08 F Spring 05. What two properties must a function T : R m R n satisfy to be a linear transformation? (a) For all vectors u and v in R m, T (u + v) T (u) + T (v)
More informationMatrix & Linear Algebra
Matrix & Linear Algebra Jamie Monogan University of Georgia For more information: http://monogan.myweb.uga.edu/teaching/mm/ Jamie Monogan (UGA) Matrix & Linear Algebra 1 / 84 Vectors Vectors Vector: A
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 informationChapter 1 Matrices and Systems of Equations
Chapter 1 Matrices and Systems of Equations System of Linear Equations 1. A linear equation in n unknowns is an equation of the form n i=1 a i x i = b where a 1,..., a n, b R and x 1,..., x n are variables.
More information1 procedure for determining the inverse matrix
table of contents 1 procedure for determining the inverse matrix The inverse matrix of a matrix A can be determined only if the determinant of the matrix A is different from zero. The following procedures
More informationMA 265 FINAL EXAM Fall 2012
MA 265 FINAL EXAM Fall 22 NAME: INSTRUCTOR S NAME:. There are a total of 25 problems. You should show work on the exam sheet, and pencil in the correct answer on the scantron. 2. No books, notes, or calculators
More informationChapter 7. Linear Algebra: Matrices, Vectors,
Chapter 7. Linear Algebra: Matrices, Vectors, Determinants. Linear Systems Linear algebra includes the theory and application of linear systems of equations, linear transformations, and eigenvalue problems.
More informationChapter 1: Systems of Linear Equations and Matrices
: Systems of Linear Equations and Matrices Multiple Choice Questions. Which of the following equations is linear? (A) x + 3x 3 + 4x 4 3 = 5 (B) 3x x + x 3 = 5 (C) 5x + 5 x x 3 = x + cos (x ) + 4x 3 = 7.
More informationMatrices and Determinants
Chapter1 Matrices and Determinants 11 INTRODUCTION Matrix means an arrangement or array Matrices (plural of matrix) were introduced by Cayley in 1860 A matrix A is rectangular array of m n numbers (or
More informationMatrices. Chapter Keywords and phrases. 3.2 Introduction
Chapter 3 Matrices 3.1 Keywords and phrases Special matrices: (row vector, column vector, zero, square, diagonal, scalar, identity, lower triangular, upper triangular, symmetric, row echelon form, reduced
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 informationLecture 2: Lattices and Bases
CSE 206A: Lattice Algorithms and Applications Spring 2007 Lecture 2: Lattices and Bases Lecturer: Daniele Micciancio Scribe: Daniele Micciancio Motivated by the many applications described in the first
More informationLecture 23: Trace and determinants! (1) (Final lecture)
Lecture 23: Trace and determinants! (1) (Final lecture) Travis Schedler Thurs, Dec 9, 2010 (version: Monday, Dec 13, 3:52 PM) Goals (2) Recall χ T (x) = (x λ 1 ) (x λ n ) = x n tr(t )x n 1 + +( 1) n det(t
More informationDeterminants. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 25
Determinants opyright c 2012 Dan Nettleton (Iowa State University) Statistics 611 1 / 25 Notation The determinant of a square matrix n n A is denoted det(a) or A. opyright c 2012 Dan Nettleton (Iowa State
More informationAPPENDIX: MATHEMATICAL INDUCTION AND OTHER FORMS OF PROOF
ELEMENTARY LINEAR ALGEBRA WORKBOOK/FOR USE WITH RON LARSON S TEXTBOOK ELEMENTARY LINEAR ALGEBRA CREATED BY SHANNON MARTIN MYERS APPENDIX: MATHEMATICAL INDUCTION AND OTHER FORMS OF PROOF When you are done
More informationDeterminant: 3.3 Properties of Determinants
Determinant: 3.3 Properties of Determinants Summer 2017 The most incomprehensible thing about the world is that it is comprehensible.  Albert Einstein Goals Learn some basic properties of determinant.
More informationLinear Algebra. Linear Equations and Matrices. Copyright 2005, W.R. Winfrey
Copyright 2005, W.R. Winfrey Topics Preliminaries Systems of Linear Equations Matrices Algebraic Properties of Matrix Operations Special Types of Matrices and Partitioned Matrices Matrix Transformations
More informationPresentation by: H. Sarper. Chapter 2  Learning Objectives
Chapter Basic Linear lgebra to accompany Introduction to Mathematical Programming Operations Research, Volume, th edition, by Wayne L. Winston and Munirpallam Venkataramanan Presentation by: H. Sarper
More informationElementary Matrices. MATH 322, Linear Algebra I. J. Robert Buchanan. Spring Department of Mathematics
Elementary Matrices MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Outline Today s discussion will focus on: elementary matrices and their properties, using elementary
More information9. The determinant. Notation: Also: A matrix, det(a) = A IR determinant of A. Calculation in the special cases n = 2 and n = 3:
9. The determinant The determinant is a function (with real numbers as values) which is defined for square matrices. It allows to make conclusions about the rank and appears in diverse theorems and formulas.
More informationDeterminants: summary of main results
Determinants: summary of main results A determinant of an n n matrix is a real number associated with this matrix. Its definition is complex for the general case We start with n = 2 and list important
More informationBasic Concepts in Linear Algebra
Basic Concepts in Linear Algebra Grady B Wright Department of Mathematics Boise State University February 2, 2015 Grady B Wright Linear Algebra Basics February 2, 2015 1 / 39 Numerical Linear Algebra Linear
More informationThe Determinant. Chapter Definition of the Determinant
Chapter 5 The Determinant 5.1 Definition of the Determinant Given a n n matrix A, we would like to define its determinant. We already have a definition for the 2 2 matrix. We define the determinant of
More information