L3-1: Determinants: The Good, the Bad and the Ugly 12 Nov 2012
|
|
- Coral West
- 6 years ago
- Views:
Transcription
1 L3-1: Determinants: The Good, the Bad and the Ugly 12 Nov 2012 Man with no name: Tut, tut. Such ingratitude after all the times I saved your life. Il Buono, il bruto, il cattivo (1966) See Hefferon, Chapter 4, p Movie trivia question #1: This film is the third and final in the Spaghetti Western Dollars Trilogy. The Man with No Name character derives from a samurai movie made only a few years earlier. What is the single most striking difference in fighting style between the samurai protagonist and the Man with No Name? And what is the MwNN s reference to a perfect number? This is for squares only: The erminant of an nxn matrix is indicated by either the symbol A or A, which is not to be confused with A -- the norm of A. Strange people also use the shorthand symbol to indicate a erminant of a given matrix. In simplest terms, the erminant is a function of the elements of an nxn matrix that yields a single number (a scalar) that, well, ermines something about the matrix: The inverse A -1 exists for matrix A if and only if A is nonzero. (In case you don t recall if and only if (iff), it means we are on a logical twoway street: IF A is nonzero, A -1 exists and if A -1 exists, A is nonzero). So if the matrix represents a set of linear equations, the erminant ermines whether the system has a unique solution. The erminant is older than mathematical dirt: first seen in the Suan shu shu, a 3 rd century BC Chinese math textbook long before the concept of a matrix was introduced. Of course, it was our old friend Gauss who first introduced the word erminant. The erminant of a matrix can be a verrry useful number if it wasn t so difficult to calculate. How we ermine the erminants A bit of slogging through the mathematical mud lies ahead. Stay with it. For the 2x2 case, the erminant is the familiar multiplication along crossed diagonals:
2 a a A A A a11a22 a12a21 a21 a22 Example: Sooo, maybe A = A T? Note: for a 2x2 matrix, the erminant formula has 2 terms, each with 2 factors. For the 3x3 case, the erminant calculation is similar, albeit considerably uglier: a11 a12 a13 A a a a A A a a a a a a a a a a31 a32 a 33 a a a a a a a a a which is something you can memorize, but not by choice. Stare at it long enough and you can maybe see the pattern (or maybe the colors help): de a a a a a a t A a11 a12 a13 a32 a33 a31 a3 3 a31 a32 Thus: The erminant of a 3x3 matrix if found by taking each entry from the first row of the original matrix and multiplying it by the erminant of a 2x2 sub-matrix. The 2x2 subs are each constructed by removing a row and a column from the original matrix, corresponding to the row, column of first row entry. Then there is that flippity-floppity minus sign. Note: 3x3 matrix, erminant formula has 6 terms, each with 3 factors. Neat trick for 3x3 s, known as Sarrus Rule: Write an augmented matrix consisting of A and the first two columns of A (yielding a 3x5). Now you can use the crossed diagonal products to find the erminant. 2
3 a a a a a a a a A a a a A a a aug 21 a22 a23 a a31 a32 a 33 a31 a3 2 a33 a32 a 33 Multiply along the down-to-the-right diagonals: A = a 11 a 22 a 33 + a 12 a 23 a 32 + a 13 a 22 a 33 but wait, there s more: Multiply along the down-to-the-left diagonals: a11 a12 a13 a11 a12 a13 a11 a12 A a a a A a a a a a aug a31 a32 a 33 a 31 a32 a33 a32 a 33 Since we are going to the left, these terms are all negative: A = a 11 a 22 a 33 + a 12 a 23 a 32 + a 23 a 22 a 33 a 13 a 22 a 31 a 11 a 23 a 32 a 12 a 22 a 33 Here is a picture of Sarrus rule in action: Solid lines + Dashed lines - That was fun, but this rule doesn t work for anything larger than 3x3. We need a formula that can generalize to any size square and one that is not a nightmare to evaluate! Yeah, there is a formula for the of a 4x4 matrix, but it has 24 terms, each with 4 factors: So what seems to be the manner in which the number of terms increases with the order of the matrix? And why is this verrry bad news for computation of erminants of large matrices? We hunt for the elusive gold -- err, a general formula for erminants: Movie trivia question #2: What is the name on the grave said to contain the missing $200,000 in gold in the movie that gives this lab its name? 3
4 Define: The minor of a matrix is a portion of the matrix with a specific row and column removed. The notation for the i,j th minor of A, where row i and column j are removed, is A ij. Example: a a a A a a A a a31 a33 a31 a32 a 33 a a The red entries in the original matrix (first row and second column) are deleted to form the minor A 12. Another way to write the 3x3 erminant is in terms of the minors: A a A A a12 12 a13 A13 This is a formula that generalizes to the nxn case: n 1 j A ( 1) a1j A1j j1 Welcome to the minor leagues (where there is no joy in Mudville) The formula for A above is further generalized to the technique known as expansion by minors or cofactor expansion. Choose any row (say row i) and form the same type of sum as above. n n i j ij ij ij ij j1 j1 i j Aij, A ( 1) a A a C where Cij ( 1) is known as the ij th cofactor of A. The cofactor is a erminant of the minor with the alternating sign (-1) i+j. The erminant may be calculated using cofactor expansion along any one row or column. Hint: pick one that has lots of 0s! Example Find Use a cofactor expansion across the first row. 4
5 ( 13) 4(10) Note the signs were ermined by (-1) 2, (-1) 3, (-1) 4. Before you celebrate finally having a formula that doesn t look too ugly, realize that this means the following: In order to calculate a erminant of a big nxn matrix, you have to calculate a lot of erminants of slightly smaller n-1 x n-1 matrices. Over and over again. Yippee! That may not or may not be ugly, but it is definitely not good. When we deal with very large matrices, this is on the order of n! flops (multiplications and additions) and that is most certainly bad! In case you wondered how this lab got its name. Find the erminant of the given matrix a Ans a. -5 b. 4 b c Properties of the erminant -- Important and useful The erminant of any identity matrix is 1. Product rule: AB A B -- actually a distributive property of the erminant operation with respect to matrix multiplication Constant multiplier: (ka) = k n A, for nxn matrix A and scalar k. Transpose: A = A T (yep, it was true when we stumbled across it above) Inverse: A = 1 / (A -1 ) ( A)( A -1 ) = AA -1 = I =1 -- also a result of the distributive property If a matrix has two equal rows (and is thus singular), its erminant is 0. 5
6 If a matrix has a row of all 0 s (and is thus singular), its erminant is 0. Do you think that generalizes (ie, singular matrices always have =0)? Question: what is the erminant of a matrix where two rows are scalar multiples? a b Suppose A What is A? Is this also true for columns that are scalar ca cb multiples? And here is what will get us out of the minors and into the bigs: The erminant of any triangular matrix is the product of the entries on the main diagonal. We love triangular matrices! So it is logical to ask: What do the row ops do to the erminant? Multiplying a row by a scalar results in multiplying the by the same scalar. Swapping any two rows yields the negative of the of the original matrix. Forming a linear combination of two rows does not change the erminant. Examples 1 3 A A scalar mult of a single row (3R 1( A) R 1): 18 3 A row swap changes sign: B B6 A linear combination, no change: C ( from B, R1R2 R2) C 6 B 6 12 This leads to a very good idea (at last!) for finding a erminant: Row reduce the square matrix A to upper triangular form via Gauss-Jordan elimination, keeping track of the row ops. Multiply the diagonal elements of the resulting triangular (we love em!) matrix. 6
7 Apply the above rules for sign changes based on scalar multiples and row swaps, if any, and you ve got your erminant. Example Reducing /2 7/2 3/2, / 5 26 / /7 requires only linear combinations of rows, so that the erminant is unchanged. The product of the diagonal entries of this upper triangular matrix is the erminant of the original matrix: -36 Practice a a. Find the erminant of A c erminant. b d. Then do a single row swap and recalculate the 3 k 4 b. Find the erminant of 2 3 k will this matrix be singular? for a nonzero constant k. For what value of k c. Find the erminant using row reduction d. Use the erminant to ermine (ha!) whether is invertible. Ans a. ad bc and cb da b. k = +/- 1 d. not invertible 7
8 Cramer s Rule: solving Ax = b with erminants (in a nicer form than you learned in Alg 2) We first need to define a column replacement operation: Let A i (b) represent the replacement of the i th column in matrix A with column vector b. Then if A is nonzero, the matrix equation Ax = b has the unique solution vector x with components: Ai ( b) xi A Named for the Swiss mathematician (and big hair guy) Gabriel Cramer, this rule is historically important, but very inefficient for problems involving large matrices. And you thought all Swiss mathematicians were named Bernoulli! Note: 80s rocker Jon BonJovi was also a big hair guy, but does not have any mathematical formulae to his name. However, his picture is on a stamp ( whilst Cramer has no stamp. What s better, being nominated for an Academy Award or finding a silly formula for erminants? Example a bx1 e a b, with 0 c d x 2 f c d e b a e f d ed bf c f af ce x1, x2 ad bc ad bc Not too bad: 14 flops to solve a 2x2 system. Less if you pre-calculate and store it (and you should do that first to be sure it is not zero). For matrices much bigger than 3x3, this process is ghastly. Cramer s rule can be used to find an inverse (but why would you?) A 1 1 C C C C C A C1n C n1 12 n2 nn, where the C ji s are cofactors. 8
9 Note the reversal of the subscripts, which normally go row, column: The i,j th entry of A -1 is formed from the j,i th cofactor of A. This C ji matrix is sometimes called the adjugate of A. Great, another made up term for another matrix. Phooey on that! Practice Find the inverse of A using the erminant. Notice anything interesting about this matrix and its inverse? Then solve Ax = b for b = 3 2. Ans x = {1, 3, -3}. 1 Answers to movie trivia questions #1: The MwNN carried only a single Colt revolver. The ronin samurai in the film Yojimbo traditionally carried two swords. MWNN: Six is the perfect number. Tuco: I thought three was the perfect number. MWNN: I ve got six bullets in my gun. Answer to movie trivia question #2: Arch Stanton. The gold wasn t actually in that grave. 9
Lemma 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 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 informationMATRICES AND MATRIX OPERATIONS
SIZE OF THE MATRIX is defined by number of rows and columns in the matrix. For the matrix that have m rows and n columns we say the size of the matrix is m x n. If matrix have the same number of rows (n)
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 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 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 informationSCHOOL OF BUSINESS, ECONOMICS AND MANAGEMENT BUSINESS MATHEMATICS / MATHEMATICAL ANALYSIS
SCHOOL OF BUSINESS, ECONOMICS AND MANAGEMENT BUSINESS MATHEMATICS / MATHEMATICAL ANALYSIS Unit Six Moses Mwale e-mail: moses.mwale@ictar.ac.zm BBA 120 Business Mathematics Contents Unit 6: Matrix Algebra
More informationFinal Review Sheet. B = (1, 1 + 3x, 1 + x 2 ) then 2 + 3x + 6x 2
Final Review Sheet The final will cover Sections Chapters 1,2,3 and 4, as well as sections 5.1-5.4, 6.1-6.2 and 7.1-7.3 from chapters 5,6 and 7. This is essentially all material covered this term. Watch
More informationCofactors and Laplace s expansion theorem
Roberto s Notes on Linear Algebra Chapter 5: Determinants Section 3 Cofactors and Laplace s expansion theorem What you need to know already: What a determinant is. How to use Gauss-Jordan elimination to
More informationA Review of Matrix Analysis
Matrix Notation Part Matrix Operations Matrices are simply rectangular arrays of quantities Each quantity in the array is called an element of the matrix and an element can be either a numerical value
More information1300 Linear Algebra and Vector Geometry
1300 Linear Algebra and Vector Geometry R. Craigen Office: MH 523 Email: craigenr@umanitoba.ca May-June 2017 Matrix Inversion Algorithm One payoff from this theorem: It gives us a way to invert matrices.
More informationL2-7 Some very stylish matrix decompositions for solving Ax = b 10 Oct 2015
L-7 Some very stylish matrix decompositions for solving Ax = b 10 Oct 015 Marty McFly: Wait a minute, Doc. Ah... Are you telling me you built a time machine... out of a DeLorean? Doc Brown: The way I see
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 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 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 informationECON 186 Class Notes: Linear Algebra
ECON 86 Class Notes: Linear Algebra Jijian Fan Jijian Fan ECON 86 / 27 Singularity and Rank As discussed previously, squareness is a necessary condition for a matrix to be nonsingular (have an inverse).
More informationFundamentals of Engineering Analysis (650163)
Philadelphia University Faculty of Engineering Communications and Electronics Engineering Fundamentals of Engineering Analysis (6563) Part Dr. Omar R Daoud Matrices: Introduction DEFINITION A matrix is
More informationExample: 2x y + 3z = 1 5y 6z = 0 x + 4z = 7. Definition: Elementary Row Operations. Example: Type I swap rows 1 and 3
Linear Algebra Row Reduced Echelon Form Techniques for solving systems of linear equations lie at the heart of linear algebra. In high school we learn to solve systems with or variables using elimination
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 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 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 information22A-2 SUMMER 2014 LECTURE 5
A- SUMMER 0 LECTURE 5 NATHANIEL GALLUP Agenda Elimination to the identity matrix Inverse matrices LU factorization Elimination to the identity matrix Previously, we have used elimination to get a system
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 informationMatrices MA1S1. Tristan McLoughlin. November 9, Anton & Rorres: Ch
Matrices MA1S1 Tristan McLoughlin November 9, 2014 Anton & Rorres: Ch 1.3-1.8 Basic matrix notation We have studied matrices as a tool for solving systems of linear equations but now we want to study them
More informationLA lecture 4: linear eq. systems, (inverses,) determinants
LA lecture 4: linear eq. systems, (inverses,) determinants Yesterday: ˆ Linear equation systems Theory Gaussian elimination To follow today: ˆ Gaussian elimination leftovers ˆ A bit about the inverse:
More informationCHAPTER 8: Matrices and Determinants
(Exercises for Chapter 8: Matrices and Determinants) E.8.1 CHAPTER 8: Matrices and Determinants (A) means refer to Part A, (B) means refer to Part B, etc. Most of these exercises can be done without a
More informationMath 313 Chapter 1 Review
Math 313 Chapter 1 Review Howard Anton, 9th Edition May 2010 Do NOT write on me! Contents 1 1.1 Introduction to Systems of Linear Equations 2 2 1.2 Gaussian Elimination 3 3 1.3 Matrices and Matrix Operations
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 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 information[Disclaimer: This is not a complete list of everything you need to know, just some of the topics that gave people difficulty.]
Math 43 Review Notes [Disclaimer: This is not a complete list of everything you need to know, just some of the topics that gave people difficulty Dot Product If v (v, v, v 3 and w (w, w, w 3, then the
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 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 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 informationMath 4A Notes. Written by Victoria Kala Last updated June 11, 2017
Math 4A Notes Written by Victoria Kala vtkala@math.ucsb.edu Last updated June 11, 2017 Systems of Linear Equations A linear equation is an equation that can be written in the form a 1 x 1 + a 2 x 2 +...
More informationMatrices. Chapter Definitions and Notations
Chapter 3 Matrices 3. Definitions and Notations Matrices are yet another mathematical object. Learning about matrices means learning what they are, how they are represented, the types of operations which
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 informationLINEAR SYSTEMS, MATRICES, AND VECTORS
ELEMENTARY LINEAR ALGEBRA WORKBOOK CREATED BY SHANNON MARTIN MYERS LINEAR SYSTEMS, MATRICES, AND VECTORS Now that I ve been teaching Linear Algebra for a few years, I thought it would be great to integrate
More informationLinear Algebra, Summer 2011, pt. 2
Linear Algebra, Summer 2, pt. 2 June 8, 2 Contents Inverses. 2 Vector Spaces. 3 2. Examples of vector spaces..................... 3 2.2 The column space......................... 6 2.3 The null space...........................
More informationMATH 341 MIDTERM 2. (a) [5 pts] Demonstrate that A and B are row equivalent by providing a sequence of row operations leading from A to B.
11/01/2011 Bormashenko MATH 341 MIDTERM 2 Show your work for all the problems. Good luck! (1) Let A and B be defined as follows: 1 1 2 A =, B = 1 2 3 0 2 ] 2 1 3 4 Name: (a) 5 pts] Demonstrate that A and
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 informationANSWERS. Answer: Perform combo(3,2,-1) on I then combo(1,3,-4) on the result. The elimination matrix is
MATH 227-2 Sample Exam 1 Spring 216 ANSWERS 1. (1 points) (a) Give a counter example or explain why it is true. If A and B are n n invertible, and C T denotes the transpose of a matrix C, then (AB 1 )
More informationMath 308 Midterm Answers and Comments July 18, Part A. Short answer questions
Math 308 Midterm Answers and Comments July 18, 2011 Part A. Short answer questions (1) Compute the determinant of the matrix a 3 3 1 1 2. 1 a 3 The determinant is 2a 2 12. Comments: Everyone seemed to
More informationMatrices and Determinants
Math Assignment Eperts is a leading provider of online Math help. Our eperts have prepared sample assignments to demonstrate the quality of solution we provide. If you are looking for mathematics help
More informationM. Matrices and Linear Algebra
M. Matrices and Linear Algebra. Matrix algebra. In section D we calculated the determinants of square arrays of numbers. Such arrays are important in mathematics and its applications; they are called matrices.
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 informationInverses and Determinants
Engineering Mathematics 1 Fall 017 Inverses and Determinants I begin finding the inverse of a matrix; namely 1 4 The inverse, if it exists, will be of the form where AA 1 I; which works out to ( 1 4 A
More informationChapter 5. Matrices. 5.1 Basic matrix notation. This material is in Chapter 1 of Anton & Rorres.
Chapter 5. Matrices This material is in Chapter of Anton & Rorres. 5. Basic matrix notation We recall that a matrix is a rectangular array or table of numbers. We call the individual numbers entries of
More informationGetting Started with Communications Engineering. Rows first, columns second. Remember that. R then C. 1
1 Rows first, columns second. Remember that. R then C. 1 A matrix is a set of real or complex numbers arranged in a rectangular array. They can be any size and shape (provided they are rectangular). A
More informationis a 3 4 matrix. It has 3 rows and 4 columns. The first row is the horizontal row [ ]
Matrices: Definition: An m n matrix, A m n is a rectangular array of numbers with m rows and n columns: a, a, a,n a, a, a,n A m,n =...... a m, a m, a m,n Each a i,j is the entry at the i th row, j th column.
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 informationSAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 Introduction to Linear Algebra
1.1. Introduction SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 Introduction to Linear algebra is a specific branch of mathematics dealing with the study of vectors, vector spaces with functions that
More informationChapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations
Chapter 1: Systems of linear equations and matrices Section 1.1: Introduction to systems of linear equations Definition: A linear equation in n variables can be expressed in the form a 1 x 1 + a 2 x 2
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 informationDeterminants - Uniqueness and Properties
Determinants - Uniqueness and Properties 2-2-2008 In order to show that there s only one determinant function on M(n, R), I m going to derive another formula for the determinant It involves permutations
More informationSAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 Introduction to Linear Algebra
SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 Introduction to 1.1. Introduction Linear algebra is a specific branch of mathematics dealing with the study of vectors, vector spaces with functions that
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 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 informationRoberto s Notes on Linear Algebra Chapter 4: Matrix Algebra Section 7. Inverse matrices
Roberto s Notes on Linear Algebra Chapter 4: Matrix Algebra Section 7 Inverse matrices What you need to know already: How to add and multiply matrices. What elementary matrices are. What you can learn
More informationDifferential Equations
This document was written and copyrighted by Paul Dawkins. Use of this document and its online version is governed by the Terms and Conditions of Use located at. The online version of this document is
More informationLinear Algebra and Matrix Inversion
Jim Lambers MAT 46/56 Spring Semester 29- Lecture 2 Notes These notes correspond to Section 63 in the text Linear Algebra and Matrix Inversion Vector Spaces and Linear Transformations Matrices are much
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 informationANSWERS (5 points) Let A be a 2 2 matrix such that A =. Compute A. 2
MATH 7- Final Exam Sample Problems Spring 7 ANSWERS ) ) ). 5 points) Let A be a matrix such that A =. Compute A. ) A = A ) = ) = ). 5 points) State ) the definition of norm, ) the Cauchy-Schwartz inequality
More informationThings we can already do with matrices. Unit II - Matrix arithmetic. Defining the matrix product. Things that fail in matrix arithmetic
Unit II - Matrix arithmetic matrix multiplication matrix inverses elementary matrices finding the inverse of a matrix determinants Unit II - Matrix arithmetic 1 Things we can already do with matrices equality
More informationChapter 4. Solving Systems of Equations. Chapter 4
Solving Systems of Equations 3 Scenarios for Solutions There are three general situations we may find ourselves in when attempting to solve systems of equations: 1 The system could have one unique solution.
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 informationL1-5. Reducing Rows 11 Aug 2014
L1-5. Reducing Rows 11 Aug 2014 Never send a human to do a machine's job. Agent Smith Primary concepts: Row-echelon form, reduced row-echelon form, Gauss-Jordan elimination method Online tetbook (Hefferon)
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 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 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 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 informationRoberto s Notes on Linear Algebra Chapter 9: Orthogonality Section 2. Orthogonal matrices
Roberto s Notes on Linear Algebra Chapter 9: Orthogonality Section 2 Orthogonal matrices What you need to know already: What orthogonal and orthonormal bases for subspaces are. What you can learn here:
More information7.5 Operations with Matrices. Copyright Cengage Learning. All rights reserved.
7.5 Operations with Matrices Copyright Cengage Learning. All rights reserved. What You Should Learn Decide whether two matrices are equal. Add and subtract matrices and multiply matrices by scalars. Multiply
More informationLinear Algebra for Beginners Open Doors to Great Careers. Richard Han
Linear Algebra for Beginners Open Doors to Great Careers Richard Han Copyright 2018 Richard Han All rights reserved. CONTENTS PREFACE... 7 1 - INTRODUCTION... 8 2 SOLVING SYSTEMS OF LINEAR EQUATIONS...
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 informationA 2. =... = c c N. 's arise from the three types of elementary row operations. If rref A = I its determinant is 1, and A = c 1
Theorem: Let A n n Then A 1 exists if and only if det A 0 proof: We already know that A 1 exists if and only if the reduced row echelon form of A is the identity matrix Now, consider reducing A to its
More information1 GSW Sets of Systems
1 Often, we have to solve a whole series of sets of simultaneous equations of the form y Ax, all of which have the same matrix A, but each of which has a different known vector y, and a different unknown
More information8-15. Stop by or call (630)
To review the basics Matrices, what they represent, and how to find sum, scalar product, product, inverse, and determinant of matrices, watch the following set of YouTube videos. They are followed by several
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 informationMath 103, Summer 2006 Determinants July 25, 2006 DETERMINANTS. 1. Some Motivation
DETERMINANTS 1. Some Motivation Today we re going to be talking about erminants. We ll see the definition in a minute, but before we get into ails I just want to give you an idea of why we care about erminants.
More informationMatrix Operations and Equations
C H A P T ER Matrix Operations and Equations 200 Carnegie Learning, Inc. Shoe stores stock various sizes and widths of each style to accommodate buyers with different shaped feet. You will use matrix operations
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 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 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 informationCMU CS 462/662 (INTRO TO COMPUTER GRAPHICS) HOMEWORK 0.0 MATH REVIEW/PREVIEW LINEAR ALGEBRA
CMU CS 462/662 (INTRO TO COMPUTER GRAPHICS) HOMEWORK 0.0 MATH REVIEW/PREVIEW LINEAR ALGEBRA Andrew ID: ljelenak August 25, 2018 This assignment reviews basic mathematical tools you will use throughout
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 informationMatrix inversion and linear equations
Learning objectives. Matri inversion and linear equations Know Cramer s rule Understand how linear equations can be represented in matri form Know how to solve linear equations using matrices and Cramer
More information1 Matrices and Systems of Linear Equations. a 1n a 2n
March 31, 2013 16-1 16. Systems of Linear Equations 1 Matrices and Systems of Linear Equations An m n matrix is an array A = (a ij ) of the form a 11 a 21 a m1 a 1n a 2n... a mn where each a ij is a real
More informationn n matrices The system of m linear equations in n variables x 1, x 2,..., x n can be written as a matrix equation by Ax = b, or in full
n n matrices Matrices Definitions Diagonal, Identity, and zero matrices Addition Multiplication Transpose and inverse The system of m linear equations in n variables x 1, x 2,..., x n a 11 x 1 + a 12 x
More informationNumerical Methods Lecture 2 Simultaneous Equations
CGN 42 - Computer Methods Numerical Methods Lecture 2 Simultaneous Equations Topics: matrix operations solving systems of equations Matrix operations: Adding / subtracting Transpose Multiplication Adding
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 informationLecture Notes in Linear Algebra
Lecture Notes in Linear Algebra Dr. Abdullah Al-Azemi Mathematics Department Kuwait University February 4, 2017 Contents 1 Linear Equations and Matrices 1 1.2 Matrices............................................
More information7.2 Matrix Algebra. DEFINITION Matrix. D 21 a 22 Á a 2n. EXAMPLE 1 Determining the Order of a Matrix d. (b) The matrix D T has order 4 * 2.
530 CHAPTER 7 Systems and Matrices 7.2 Matrix Algebra What you ll learn about Matrices Matrix Addition and Subtraction Matrix Multiplication Identity and Inverse Matrices Determinant of a Square Matrix
More information. =. a i1 x 1 + a i2 x 2 + a in x n = b i. a 11 a 12 a 1n a 21 a 22 a 1n. i1 a i2 a in
Vectors and Matrices Continued Remember that our goal is to write a system of algebraic equations as a matrix equation. Suppose we have the n linear algebraic equations a x + a 2 x 2 + a n x n = b a 2
More informationNumerical Methods Lecture 2 Simultaneous Equations
Numerical Methods Lecture 2 Simultaneous Equations Topics: matrix operations solving systems of equations pages 58-62 are a repeat of matrix notes. New material begins on page 63. Matrix operations: Mathcad
More information6.4 Determinants and Cramer s Rule
6.4 Determinants and Cramer s Rule Objectives Determinant of a 2 x 2 Matrix Determinant of an 3 x 3 Matrix Determinant of a n x n Matrix Cramer s Rule If a matrix is square (that is, if it has the same
More informationEXAM 2 REVIEW DAVID SEAL
EXAM 2 REVIEW DAVID SEAL 3. Linear Systems and Matrices 3.2. Matrices and Gaussian Elimination. At this point in the course, you all have had plenty of practice with Gaussian Elimination. Be able to row
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 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 informationA Brief Outline of Math 355
A Brief Outline of Math 355 Lecture 1 The geometry of linear equations; elimination with matrices A system of m linear equations with n unknowns can be thought of geometrically as m hyperplanes intersecting
More information