Chapter 2:Determinants. Section 2.1: Determinants by cofactor expansion

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Chapter 2:Determinants. Section 2.1: Determinants by cofactor expansion"

Transcription

1 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 c d a real number with a square matrix. In this chapter, we will learn how to calculate the determinant of n n matrices. Definition: If A is a square matrix of order n, then the minor of entry a ij, denoted M ij, is the determinant of the n 1 n 1 submatrix obtained by removing the ith row and jth column from A. The cofactor of entry a ij is C ij = ( 1) i+j M ij. Example Note: C ij = ±M ij so to find the cofactor you only need to determine the sign once you know the minor entry. 1

2 Theorem 2.1.1: Expansions by cofactors The determinant of a square matrix of order n can be calculated by multiplying any row or column by their cofactors and adding the products. That is, using the cofactor expansion along row i: or along column j: det(a) = a i1 C i1 + a i2 C i2 + + a in C in det(a) = a 1j C 1j + a 2j C 2j + + a nj C nj Note: Since we can choose any row or column for the cofactor expansion, choosing the row or column with the most zeros will cut down on the number of calculations. Note: Choosing any other row or column would give the same answer. 2

3 Definition: If A is a square matrix of order n and C ij is the cofactor of entry a ij, then the matrix of cofactors from A has the form C 11 C 12 C 1n C 21 C 22 C 2n.. C n1 C n2 C nn The transpose of this matrix is called the adjoint of A. That is, adj(a) = = C 11 C 12 C 1n C 21 C 22 C 2n T.. C n1 C n2 C nn C 11 C 21 C n1 C 12 C 22 C n2.. C 1n C 2n C nn 3

4 Theorem 2.1.2: Inverse of a matrix using its adjoint If A is an invertible matrix, then A 1 1 = det(a) adj(a) Definition: A square matrix of order n is upper triangular is all entries below the main diagonal are zero, lower triangular if all entries above the main diagonal are zero, and diagonal is all entries off the main diagonal are zero. Examples: Theorem 2.1.3: If A is a square matrix of order n and is upper triangular, lower triangular or diagonal then det(a) is the product of the entries 4

5 on the main diagonal. That is, Proof: det(a) = a 11 a 22 a nn Examples: 5

6 Theorem 1.7.1: 1. The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. 2. The product of lower triangular matrices is lower triangular, and the product of upper triangular matrices is upper triangular 3. A triangular matrix is invertible if and only if its diagonal entries are all nonzero 4. The inverse of an invertible lower triangular matrix is lower triangular, and the inverse of an invertible upper triangular matrix is upper triangular Proof: See text; uses the adjoint Theorem Cramer s rule: If A x = b is a system of n linear equations and det(a) 0 (that is, A is invertible) then the system has a unique solution given by x 1 = det(a 1) det(a), x 2 = det(a 2) det(a),...,x n = det(a n) det(a) where A j is the matrix obtained by replacing the jth column of A with the vector matrix b. 6

7 Section 2.2: Evaluating determinants by row reduction Theorem 2.2.1: If A is a square matrix of order n and has a row or column of zeros, then det(a) = 0. Proof: Theorem 2.2.2: Let A be a square matrix of order n, then det(a) = det(a T ). Proof: Note: This means almost any statement about determinants that talks about rows is also true for columns. Theorem 2.2.3: Let A be a square matrix of order n 1. If B is the matrix obtained by multiplying a single row or column of A by the scalar k (scaling), then det(b) = k det(a) 2. If B is the matrix obtained by interchanging two rows or columns of A, then det(b) = det(a) 3. If B is the matrix obtained when one row or column of A is added to a multiple of another row or column (replacement), then det(b) = det(a) We can make the same statements about elementary matrices. Theorem 2.2.4: Let E be an elementary square matrix of order n. 1. If E is obtained by a multiplying a row of I n by k, then det(e) = k 2. If E is obtained by interchanging two rows of I n, then det(e) = 1 3. If E results from adding a multiple of one row of I n to another, then det(e) = 1 7

8 Theorem 2.2.5: If A is a square matrix of order n with two proportional rows or columns, then det(a) = 0 Examples: 8

9 There are often ways we can make the calculation of the determinant a little easier. Idea 1: Use elementary row operations to reduce a given matrix to a triangular matrix Note: Be very careful not to use scaling and replacement at the same time (eg replace row1 by row1 + k row2, not by a row1 + b row2) 9

10 Idea 2: Use elementary row operations to produce rows or columns with only one non-zero entry then use cofactor expansion. 10

11 Section 2.3: Properties of the determinant Basic properties of the determinant: 1. det(ka) = k n det(a), k scalar 2. det(a + B) det(a) + det(b) 3. If A, B, C are square matrices of order n that differ only in the rth row, and the rth row of C can obtained by adding the corresponding entries in the rth row if A and B, then det(c) = det(a) + det(b) (Theorem 2.3.1) 4. If E is an elementary matrix, then det(eb) = det(e)det(b) (Lemma 2.3.2) 5. A square matrix is invertible if and only if det(a) 0 (Theorem 2.3.3) 6. If A and B are square matrices of order n, then det(ab) = det(a)det(b) (Theorem 2.3.4) 7. If A is invertible, then det(a 1 ) = 1 det(a) Examples: (Theorem 2.3.5) 11

12 Theorem 2.3.6: If A is a square matrix of order n, then the following statements are equivalent 1. A is invertible 2. A x = 0 has only the trivial solution 3. The reduced row echelon form of A is I n 4. A can be expressed as a product of elementary matrices 5. A x = b is consistent for every n 1 matrix b 6. A x = b has exactly one solution for every n 1 matrix b 7. det(a) 0 Often we need to solve problems of the form A x = λ x where λ is a scalar. This is a system of n linear equations. Definition: The value(s) of λ for which the system of equations (λi A) x = 0 has a nontrivial solution of called the characteristic value or eigenvalue of A. If λ is an eigenvalue of A, then the nontrivial solutions of the system are called the eigenvectors of A corresponding to λ. Note: Eigenvalues and eigenvectors are used alot, especially when modelling and analyzing physical systems. 12

Determinants by Cofactor Expansion (III)

Determinants 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 information

Evaluating Determinants by Row Reduction

Evaluating 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 information

Chapter 4. Determinants

Chapter 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 information

Determinants Chapter 3 of Lay

Determinants 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 information

Lecture 10: Determinants and Cramer s Rule

Lecture 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 information

c c c c c c c c c c a 3x3 matrix C= has a determinant determined by

c 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 information

4. Determinants.

4. 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 information

Formula for the inverse matrix. Cramer s rule. Review: 3 3 determinants can be computed expanding by any row or column

Formula 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 information

Determinants. 2.1 Determinants by Cofactor Expansion. Recall from Theorem that the 2 2 matrix

Determinants. 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 information

Math 240 Calculus III

Math 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 information

Determinants. Samy Tindel. Purdue University. Differential equations and linear algebra - MA 262

Determinants. 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 information

Introduction to Determinants

Introduction 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 information

det(ka) = k n det A.

det(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 information

Math Linear Algebra Final Exam Review Sheet

Math 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 information

k=1 ( 1)k+j M kj detm kj. detm = ad bc. = 1 ( ) 2 ( )+3 ( ) = = 0

k=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 information

and let s calculate the image of some vectors under the transformation T.

and 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 information

1 Determinants. 1.1 Determinant

1 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 information

Determinants. Beifang Chen

Determinants. 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 information

MATH 1210 Assignment 4 Solutions 16R-T1

MATH 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 information

Determinants and Scalar Multiplication

Determinants 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 information

Determinants. 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 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 information

ENGR-1100 Introduction to Engineering Analysis. Lecture 21

ENGR-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 information

TOPIC III LINEAR ALGEBRA

TOPIC 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 information

Math Camp Notes: Linear Algebra I

Math 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 information

Linear Algebra and Vector Analysis MATH 1120

Linear 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 information

Calculating determinants for larger matrices

Calculating 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 information

Announcements Wednesday, October 25

Announcements 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 information

LECTURE 4: DETERMINANT (CHAPTER 2 IN THE BOOK)

LECTURE 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 information

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET

IMPORTANT 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 information

Topic 15 Notes Jeremy Orloff

Topic 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 information

MTH 102A - Linear Algebra II Semester

MTH 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 information

MAC Module 2 Systems of Linear Equations and Matrices II. Learning Objectives. Upon completing this module, you should be able to :

MAC 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 information

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET

IMPORTANT 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 information

A 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

A 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 information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2

MATRIX 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 information

Math 110 Linear Algebra Midterm 2 Review October 28, 2017

Math 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 information

Linear 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 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 information

Introduction to Matrices

Introduction to Matrices 214 Analysis and Design of Feedback Control Systems Introduction to Matrices Derek Rowell October 2002 Modern system dynamics is based upon a matrix representation of the dynamic equations governing the

More information

Matrix Operations: Determinant

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 information

The Determinant: a Means to Calculate Volume

The 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 information

I = i 0,

I = 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 information

Linear Algebra: Sample Questions for Exam 2

Linear Algebra: Sample Questions for Exam 2 Linear Algebra: Sample Questions for Exam 2 Instructions: This is not a comprehensive review: there are concepts you need to know that are not included. Be sure you study all the sections of the book and

More information

MATH 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. 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 information

Linear Algebra Primer

Linear 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 information

Chapters 5 & 6: Theory Review: Solutions Math 308 F Spring 2015

Chapters 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 information

Determinants An Introduction

Determinants 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 information

ELEMENTARY LINEAR ALGEBRA WITH APPLICATIONS. 1. Linear Equations and Matrices

ELEMENTARY 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 information

MATH 323 Linear Algebra Lecture 6: Matrix algebra (continued). Determinants.

MATH 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 information

Chapter 2: Matrices and Linear Systems

Chapter 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 information

Presentation by: H. Sarper. Chapter 2 - Learning Objectives

Presentation 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 information

Math 416, Spring 2010 The algebra of determinants March 16, 2010 THE ALGEBRA OF DETERMINANTS. 1. Determinants

Math 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 information

Notes on Determinants and Matrix Inverse

Notes 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 information

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition

Linear 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 information

Determinants: summary of main results

Determinants: 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 information

Determinants of 2 2 Matrices

Determinants 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 information

1 procedure for determining the inverse matrix

1 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 information

MATRICES. knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns.

MATRICES. knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns. MATRICES After studying this chapter you will acquire the skills in knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns. List of

More information

1. General Vector Spaces

1. General Vector Spaces 1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule

More information

Determinant: 3.3 Properties of Determinants

Determinant: 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 information

A matrix A is invertible i det(a) 6= 0.

A matrix A is invertible i det(a) 6= 0. Chapter 4 Determinants 4.1 Definition Using Expansion by Minors Every square matrix A has a number associated to it and called its determinant, denotedbydet(a). One of the most important properties of

More information

The Determinant. Chapter Definition of the Determinant

The 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

MTH50 Spring 07 HW Assignment 7 {From [FIS0]}: Sec 44 #4a h 6; Sec 5 #ad ac 4ae 4 7 The due date for this assignment is 04/05/7 Sec 44 #4a h Evaluate the erminant of the following matrices by any legitimate

More information

Determinants and Scalar Multiplication

Determinants 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 information

Fall Inverse of a matrix. Institute: UC San Diego. Authors: Alexander Knop

Fall Inverse of a matrix. Institute: UC San Diego. Authors: Alexander Knop Fall 2017 Inverse of a matrix Authors: Alexander Knop Institute: UC San Diego Row-Column Rule If the product AB is defined, then the entry in row i and column j of AB is the sum of the products of corresponding

More information

Fundamentals of Linear Algebra. Marcel B. Finan Arkansas Tech University c All Rights Reserved

Fundamentals of Linear Algebra. Marcel B. Finan Arkansas Tech University c All Rights Reserved Fundamentals of Linear Algebra Marcel B. Finan Arkansas Tech University c All Rights Reserved 2 PREFACE Linear algebra has evolved as a branch of mathematics with wide range of applications to the natural

More information

4. Linear transformations as a vector space 17

4. Linear transformations as a vector space 17 4 Linear transformations as a vector space 17 d) 1 2 0 0 1 2 0 0 1 0 0 0 1 2 3 4 32 Let a linear transformation in R 2 be the reflection in the line = x 2 Find its matrix 33 For each linear transformation

More information

= 1 and 2 1. T =, and so det A b d

= 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 information

Math 215 HW #9 Solutions

Math 215 HW #9 Solutions Math 5 HW #9 Solutions. Problem 4.4.. If A is a 5 by 5 matrix with all a ij, then det A. Volumes or the big formula or pivots should give some upper bound on the determinant. Answer: Let v i be the ith

More information

Lecture 6 & 7. Shuanglin Shao. September 16th and 18th, 2013

Lecture 6 & 7. Shuanglin Shao. September 16th and 18th, 2013 Lecture 6 & 7 Shuanglin Shao September 16th and 18th, 2013 1 Elementary matrices 2 Equivalence Theorem 3 A method of inverting matrices Def An n n matrice is called an elementary matrix if it can be obtained

More information

Math 18, Linear Algebra, Lecture C00, Spring 2017 Review and Practice Problems for Final Exam

Math 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 information

SPRING OF 2008 D. DETERMINANTS

SPRING 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 information

1. Select the unique answer (choice) for each problem. Write only the answer.

1. Select the unique answer (choice) for each problem. Write only the answer. MATH 5 Practice Problem Set Spring 7. Select the unique answer (choice) for each problem. Write only the answer. () Determine all the values of a for which the system has infinitely many solutions: x +

More information

THE 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.

THE 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 information

Math Lecture 26 : The Properties of Determinants

Math 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 information

Matrices. Ellen Kulinsky

Matrices. Ellen Kulinsky Matrices Ellen Kulinsky Amusement Parks At an amusement park, each adult ticket costs $10 and each children s ticket costs $5. At the end of one day, the amusement park as sold $200 worth of tickets. You

More information

Math 313 (Linear Algebra) Exam 2 - Practice Exam

Math 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 information

Math Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88

Math 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

Chapter 2. Determinants

Chapter 2. Determinants Chpter Determinnts The Determinnt Function Recll tht the X mtrix A c b d is invertible if d-bc0. The expression d-bc occurs so frequently tht it hs nme; it is clled the determinnt of the mtrix A nd is

More information

CHAPTER 6. Direct Methods for Solving Linear Systems

CHAPTER 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 information

Elementary matrices, continued. To summarize, we have identified 3 types of row operations and their corresponding

Elementary matrices, continued. To summarize, we have identified 3 types of row operations and their corresponding Elementary matrices, continued To summarize, we have identified 3 types of row operations and their corresponding elementary matrices. If you check the previous examples, you ll find that these matrices

More information

The Matrix-Tree Theorem

The Matrix-Tree Theorem The Matrix-Tree 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 Matrix-Tree Theorem. 1 Preliminaries

More information

MATH 369 Linear Algebra

MATH 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

UNIT 3 MATRICES - II

UNIT 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 information

0.1 Eigenvalues and Eigenvectors

0.1 Eigenvalues and Eigenvectors 0.. EIGENVALUES AND EIGENVECTORS MATH 22AL Computer LAB for Linear Algebra Eigenvalues and Eigenvectors Dr. Daddel Please save your MATLAB Session (diary)as LAB9.text and submit. 0. Eigenvalues and Eigenvectors

More information

Section 5.3 Systems of Linear Equations: Determinants

Section 5.3 Systems of Linear Equations: Determinants Section 5. Systems of Linear Equations: Determinants In this section, we will explore another technique for solving systems called Cramer's Rule. Cramer's rule can only be used if the number of equations

More information

Foundations of Cryptography

Foundations 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 information

CHAPTER 8: Matrices and Determinants

CHAPTER 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 information

7.4. The Inverse of a Matrix. Introduction. Prerequisites. Learning Outcomes

7.4. 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 information

More chapter 3...linear dependence and independence... vectors

More chapter 3...linear dependence and independence... vectors More chapter 3...linear dependence and independence... vectors It is important to determine if a set of vectors is linearly dependent or independent Consider a set of vectors A, B, and C. If we can find

More information

Determinants: Elementary Row/Column Operations

Determinants: 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 information

a11 a A = : a 21 a 22

a11 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 information

Systems of Linear Equations. By: Tri Atmojo Kusmayadi and Mardiyana Mathematics Education Sebelas Maret University

Systems of Linear Equations. By: Tri Atmojo Kusmayadi and Mardiyana Mathematics Education Sebelas Maret University Systems of Linear Equations By: Tri Atmojo Kusmayadi and Mardiyana Mathematics Education Sebelas Maret University Standard of Competency: Understanding the properties of systems of linear equations, matrices,

More information

Matrix Algebra & Elementary Matrices

Matrix Algebra & Elementary Matrices Matrix lgebra & Elementary Matrices To add two matrices, they must have identical dimensions. To multiply them the number of columns of the first must equal the number of rows of the second. The laws below

More information

ft-uiowa-math2550 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 12/31/2014 at 10:36pm CST

ft-uiowa-math2550 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 12/31/2014 at 10:36pm CST me me ft-uiowa-math255 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 2/3/2 at :3pm CST. ( pt) Library/TCNJ/TCNJ LinearSystems/problem3.pg Give a geometric description of the following

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 =

(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 information

14. Properties of the Determinant

14. Properties of the Determinant 14. Properties of the Determinant Last time we showed that the erminant of a matrix is non-zero if and only if that matrix is invertible. We also showed that the erminant is a multiplicative function,

More information

18.06SC Final Exam Solutions

18.06SC Final Exam Solutions 18.06SC Final Exam Solutions 1 (4+7=11 pts.) Suppose A is 3 by 4, and Ax = 0 has exactly 2 special solutions: 1 2 x 1 = 1 and x 2 = 1 1 0 0 1 (a) Remembering that A is 3 by 4, find its row reduced echelon

More information

1 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 )

1 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 Per-Olof Persson persson@berkeleyedu Department of Mathematics University of California, Berkeley Math 18A Numerical

More information

Matrices and Determinants

Matrices 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 information

Math 225 Linear Algebra II Lecture Notes. John C. Bowman University of Alberta Edmonton, Canada

Math 225 Linear Algebra II Lecture Notes. John C. Bowman University of Alberta Edmonton, Canada Math 225 Linear Algebra II Lecture Notes John C Bowman University of Alberta Edmonton, Canada March 23, 2017 c 2010 John C Bowman ALL RIGHTS RESERVED Reproduction of these lecture notes in any form, in

More information