DETERMINANTS, MINORS AND COFACTORS ALGEBRA 6. Dr Adrian Jannetta MIMA CMath FRAS INU0114/514 (MATHS 1)
|
|
- Jacob Crawford
- 6 years ago
- Views:
Transcription
1 DETERMINANTS, MINORS AND COFACTORS ALGEBRA 6 INU0114/514 (MATHS 1) Dr Adrian Jannetta MIMA CMath FRAS Determinants, minors and cofactors 1/ 15 Adrian Jannetta
2 Determinants Every square matrix has a number called a determinant associated with it. The determinant is calculated using the elements of the matrix. Determinants have many applications in mathematics. They are useful for deciding whether a set of equations have consistent solutions (for example, in deciding whether or not vector lines are skew or intersecting). As you will see, the determinant is also necessary for calculating the inverse of a matrix. In this presentation we ll only consider determinants of 2 2 and 3 3 matrices. Determinants, minors and cofactors 2/ 15 Adrian Jannetta
3 Second order determinants The 2 2 matrix a b A= c d has a determinant which is calculated as follows: A = a b c d = ad bc Example Find the determinant of the matrix B= In this case the determinant is B = =(3 4) (1 2)=12+2=14. Determinants, minors and cofactors 3/ 15 Adrian Jannetta
4 Third order determinants Determinants of of matrices of size 3 3 (or larger) are evaluated differently. To calculate them we must include the appropriate sign (±) for each element. The signs of elements alternate between plus and minus like this: a b c + + d e f elements associated with signs + g h i + + A plus sign means leave the element unchanged. A minus means flip the sign of the element. The 3 3 determinant is first expressed in terms of the 2 2 determinants. Determinants, minors and cofactors 4/ 15 Adrian Jannetta
5 Minors Each element of a 3 3 matrix is associated with a 2 2 determinant called a minor. It is found by taking elements from the row and column not containing that element. Consider the matrix a b c d e f g h i a b c d e f g h i a b c d e f g h i a b c d e f g h i a b c d e f g h i Element a is associated with the 2 2 minor e h Element b is associated with the 2 2 minor d g Element c is associated with the 2 2 minor d g Element d is associated with the 2 2 minor b h f i. f i. e h. c i. Determinants,...and sominors on. and cofactors 5/ 15 Adrian Jannetta
6 3 3 determinants can be expanded along any row or column. For simplicity, let s see how to expand along the first row. We note that the signs are+ + in the expansion of the first row. a b c d e f g h i = a e h f i b d g f i + c d g e h Now we can expand each of the 2 2 determinants in the usual way: a b c d e f = a(ei fh) b(di fg)+c(dh eg) g h i This is not a formula to memorise try to learn the method of expansion. Let s try this on a matrix with actual values. Determinants, minors and cofactors 6/ 15 Adrian Jannetta
7 Evaluating a 3 3 determinant Find the determinant associated with the matrix A= Let us expand along the first row: deta = = deta = 2(42 12) 5(54 0)+1(27 0) = 2(30) 5(54)+1(27) = = 183 Determinants, minors and cofactors 7/ 15 Adrian Jannetta
8 Evaluating a 3 3 determinant Evaluate If we expand along the first row: = = 3(0)+2( 32)+8 = 64+8= 56 ( 2) However, it is quicker to use the 3rd row (with those zeros!) = = 8( 7) = 56 Determinants, minors and cofactors 8/ 15 Adrian Jannetta
9 Cofactor matrix Given the matrix A= a b c d e f g h i we saw previously that each element is associated with a 2 2 determinant. The values of those determinants are called cofactors. If we calculate the values of all 9 determinants, taking into account the behaviour of a b c + + d e f elements associated with signs + g h i + + we can make a matrix from the cofactors.remember: a plus means leave the element unchanged. A minus means flip the sign of the element. Calculating the cofactor matrix will be an important step in calculating inverse of a matrix in the next presentation. Determinants, minors and cofactors 9/ 15 Adrian Jannetta
10 Cofactor matrix Given A= find the matrix of cofactors C Let s work along the top row of A to calculate the elements of the new matrix C. Remember: + leaves the sign unchanged, while flips it. C 11 = =+( 20 2)= 22 C 12 = = (5 2)= 3 C 13 = =+( 2 8)= 10 So far the cofactor matrix looks like this: C= We ll continue with the second row. Determinants, minors and cofactors 10/ 15 Adrian Jannetta
11 C 21 = C 22 =+ C 23 = = ( 15+4)=11 =+( 5+4)= 1 = (2 6)=4 Now the cofactor matrix looks like this: C= And now the final row... Determinants, minors and cofactors 11/ 15 Adrian Jannetta
12 C 31 =+ C 32 = C 33 = =+(3+8)=11 = (1 2)=1 =+(4+3)=7 The completed matrix of cofactors is C= The cofactor matrix is a crucial step in finding the inverse of a matrix later in the next presentation. Determinants, minors and cofactors 12/ 15 Adrian Jannetta
13 Singular and nonsingular matrices The determinant provides important information about the matrix. A matrix whose determinant is zero is said to be singular. A matrix whose determinant is not zero is said to be nonsingular. For example, the equations 2x+3y = 10 4x 5y = 16 can be represented in matrix form by 2 3 x 4 5 y = The determinant of the matrix of coefficients tells us whether or not there are solutions to original set of equations. Here, we have A= det A= 22 Since A is nonsingular then the two equations do have solutions. If A had been singular (deta = 0) then it would not have solutions. Determinants, minors and cofactors 13/ 15 Adrian Jannetta We will soon be using matrices to solve systems of equations and the
14 Solutions of linear systems Determine whether or not the equations 2.5x+13y = x 182y = have a unique solution. In matrix form the equations are Ax=b: x y = The matrix of coefficients will tell us about the solution the RHS of the equation is unimportant A= The determinant is det A=( 2.5)( 182) (13)(35)= =0 Therefore, there are no unique solutions to the original equations. Determinants, minors and cofactors 14/ 15 Adrian Jannetta
15 Test yourself If you ve understood the ideas and examples presented in these notes then you should be able to solve the following problems Given the matrices A=, B= and C= Evaluate det A. 2 Show that B is a singular matrix. 3 Find detc. 4 Write down the cofactor matrix for C. 1 det A=26. 2 det B = 0, therefore B is singular. 3 det C=83. 4 Cofactor matrix Determinants, minors and cofactors 15/ 15 Adrian Jannetta
REVERSE CHAIN RULE CALCULUS 7. Dr Adrian Jannetta MIMA CMath FRAS INU0115/515 (MATHS 2) Reverse Chain Rule 1/12 Adrian Jannetta
REVERSE CHAIN RULE CALCULUS 7 INU05/55 (MATHS 2) Dr Adrian Jannetta MIMA CMath FRAS Reverse Chain Rule /2 Adrian Jannetta Reversing the chain rule In differentiation the chain rule is used to get the derivative
More informationBASIC ALGEBRA ALGEBRA 1. Dr Adrian Jannetta MIMA CMath FRAS INU0114/514 (MATHS 1) Basic algebra 1/ 17 Adrian Jannetta
BASIC ALGEBRA ALGEBRA 1 INU0114/514 (MATHS 1) Dr Adrian Jannetta MIMA CMath FRAS Basic algebra 1/ 17 Adrian Jannetta Overview In this presentation we will review some basic definitions and skills required
More informationQUADRATIC GRAPHS ALGEBRA 2. Dr Adrian Jannetta MIMA CMath FRAS INU0114/514 (MATHS 1) Quadratic Graphs 1/ 16 Adrian Jannetta
QUADRATIC GRAPHS ALGEBRA 2 INU0114/514 (MATHS 1) Dr Adrian Jannetta MIMA CMath FRAS Quadratic Graphs 1/ 16 Adrian Jannetta Objectives Be able to sketch the graph of a quadratic function Recognise the shape
More informationIntroduction Derivation General formula List of series Convergence Applications Test SERIES 4 INU0114/514 (MATHS 1)
MACLAURIN SERIES SERIES 4 INU0114/514 (MATHS 1) Dr Adrian Jannetta MIMA CMath FRAS Maclaurin Series 1/ 21 Adrian Jannetta Recap: Binomial Series Recall that some functions can be rewritten as a power series
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 informationRATIONAL FUNCTIONS AND
RATIONAL FUNCTIONS AND GRAPHS ALGEBRA 5 INU0114/514 (MATHS 1) Dr Adrian Jannetta MIMA CMath FRAS Rational functions and graphs 1/ 20 Adrian Jannetta Objectives In this lecture (and next seminar) we will
More informationIntroduction Derivation General formula Example 1 List of series Convergence Applications Test SERIES 4 INU0114/514 (MATHS 1)
MACLAURIN SERIES SERIES 4 INU0114/514 (MATHS 1) Dr Adrian Jannetta MIMA CMath FRAS Maclaurin Series 1/ 19 Adrian Jannetta Background In this presentation you will be introduced to the concept of a power
More informationMath 240 Calculus III
The Calculus III Summer 2015, Session II Wednesday, July 8, 2015 Agenda 1. of the determinant 2. determinants 3. of determinants What is the determinant? Yesterday: Ax = b has a unique solution when A
More 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 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 informationPRODUCT & QUOTIENT RULES CALCULUS 2. Dr Adrian Jannetta MIMA CMath FRAS INU0115/515 (MATHS 2) Product & quotient rules 1/13 Adrian Jannetta
PRODUCT & QUOTIENT RULES CALCULUS 2 INU0115/515 (MATHS 2) Dr Adrian Jannetta MIMA CMath FRAS Proct & quotient rules 1/13 Adrian Jannetta Objectives In this presentation we ll continue learning how to differentiate
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 informationMATRIX DETERMINANTS. 1 Reminder Definition and components of a matrix
MATRIX DETERMINANTS Summary Uses... 1 1 Reminder Definition and components of a matrix... 1 2 The matrix determinant... 2 3 Calculation of the determinant for a matrix... 2 4 Exercise... 3 5 Definition
More informationCOMPLEX NUMBERS ALGEBRA 7. Dr Adrian Jannetta MIMA CMath FRAS INU0114/514 (MATHS 1) Complex Numbers 1/ 22 Adrian Jannetta
COMPLEX NUMBERS ALGEBRA 7 INU0114/514 (MATHS 1) Dr Adrian Jannetta MIMA CMath FRAS Complex Numbers 1/ 22 Adrian Jannetta Objectives This presentation will cover the following: Introduction to complex numbers.
More informationMATH 2050 Assignment 8 Fall [10] 1. Find the determinant by reducing to triangular form for the following matrices.
MATH 2050 Assignment 8 Fall 2016 [10] 1. Find the determinant by reducing to triangular form for the following matrices. 0 1 2 (a) A = 2 1 4. ANS: We perform the Gaussian Elimination on A by the following
More informationMath 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 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 informationIMPLICIT DIFFERENTIATION
IMPLICIT DIFFERENTIATION CALCULUS 3 INU0115/515 (MATHS 2) Dr Arian Jannetta MIMA CMath FRAS Implicit Differentiation 1/ 11 Arian Jannetta Explicit an implicit functions Explicit functions An explicit function
More informationSMALL INCREMENTS CALCULUS 13. Dr Adrian Jannetta MIMA CMath FRAS INU0115/515 (MATHS 2) Small increments 1/15 Adrian Jannetta
SMALL INCREMENTS CALCULUS 13 INU0115/515 (MATHS 2) Dr Adrian Jannetta MIMA CMath FRAS Small increments 1/15 Adrian Jannetta Objectives In this presentation we re going to look at one of the applications
More informationMAT 1332: CALCULUS FOR LIFE SCIENCES. Contents. 1. Review: Linear Algebra II Vectors and matrices Definition. 1.2.
MAT 1332: CALCULUS FOR LIFE SCIENCES JING LI Contents 1 Review: Linear Algebra II Vectors and matrices 1 11 Definition 1 12 Operations 1 2 Linear Algebra III Inverses and Determinants 1 21 Inverse Matrices
More informationDeterminants: Uniqueness and more
Math 5327 Spring 2018 Determinants: Uniqueness and more Uniqueness The main theorem we are after: Theorem 1 The determinant of and n n matrix A is the unique n-linear, alternating function from F n n to
More 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 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 information7.3. Determinants. Introduction. Prerequisites. Learning Outcomes
Determinants 7.3 Introduction Among other uses, determinants allow us to determine whether a system of linear equations has a unique solution or not. The evaluation of a determinant is a key skill in engineering
More informationPARTIAL DIFFERENTIATION
PARTIAL DIFFERENTIATION CALCULUS 13 INU0115/515 (MATHS 2) Dr Adrian Jannetta MIMA CMath FRAS Partial Differentiation 1/ 15 Adrian Jannetta Functions of many variables The functions and derivatives which
More informationLinear algebra and differential equations (Math 54): Lecture 7
Linear algebra and differential equations (Math 54): Lecture 7 Vivek Shende February 9, 2016 Hello and welcome to class! Last time We introduced linear subspaces and bases. Today We study the determinant
More informationMath 2331 Linear Algebra
2.2 The Inverse of a Matrix Math 2331 Linear Algebra 2.2 The Inverse of a Matrix Shang-Huan Chiu Department of Mathematics, University of Houston schiu@math.uh.edu math.uh.edu/ schiu/ Shang-Huan Chiu,
More informationDiagonalization. MATH 1502 Calculus II Notes. November 4, 2008
Diagonalization MATH 1502 Calculus II Notes November 4, 2008 We want to understand all linear transformations L : R n R m. The basic case is the one in which n = m. That is to say, the case in which the
More informationLinear algebra and differential equations (Math 54): Lecture 8
Linear algebra and differential equations (Math 54): Lecture 8 Vivek Shende February 11, 2016 Hello and welcome to class! Last time We studied the formal properties of determinants, and how to compute
More informationMH1200 Final 2014/2015
MH200 Final 204/205 November 22, 204 QUESTION. (20 marks) Let where a R. A = 2 3 4, B = 2 3 4, 3 6 a 3 6 0. For what values of a is A singular? 2. What is the minimum value of the rank of A over all a
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 informationMATRICES The numbers or letters in any given matrix are called its entries or elements
MATRICES A matrix is defined as a rectangular array of numbers. Examples are: 1 2 4 a b 1 4 5 A : B : C 0 1 3 c b 1 6 2 2 5 8 The numbers or letters in any given matrix are called its entries or elements
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 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 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 informationComponents and change of basis
Math 20F Linear Algebra Lecture 16 1 Components and change of basis Slide 1 Review: Isomorphism Review: Components in a basis Unique representation in a basis Change of basis Review: Isomorphism Definition
More 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 informationD. Determinants. a b c d e f
D. Determinants Given a square array of numers, we associate with it a numer called the determinant of, and written either det(), or. For 2 2 and 3 3 arrays, the numer is defined y () a c d = ad c; a c
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 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 informationFebruary 20 Math 3260 sec. 56 Spring 2018
February 20 Math 3260 sec. 56 Spring 2018 Section 2.2: Inverse of a Matrix Consider the scalar equation ax = b. Provided a 0, we can solve this explicity x = a 1 b where a 1 is the unique number such that
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 informationMath Computation Test 1 September 26 th, 2016 Debate: Computation vs. Theory Whatever wins, it ll be Huuuge!
Math 5- Computation Test September 6 th, 6 Debate: Computation vs. Theory Whatever wins, it ll be Huuuge! Name: Answer Key: Making Math Great Again Be sure to show your work!. (8 points) Consider the following
More informationAREA UNDER A CURVE CALCULUS 8. Dr Adrian Jannetta MIMA CMath FRAS INU0115/515 (MATHS 2) Area under a curve 1/15 Adrian Jannetta
AREA UNDER A CURVE CALCULUS 8 INU0115/515 (MATHS 2) Dr Adrian Jannetta MIMA CMath FRAS Area under a curve 1/15 Adrian Jannetta The area beneath a curve =f() The total area beneath the curve is approimatel
More informationMaterials engineering Collage \\ Ceramic & construction materials department Numerical Analysis \\Third stage by \\ Dalya Hekmat
Materials engineering Collage \\ Ceramic & construction materials department Numerical Analysis \\Third stage by \\ Dalya Hekmat Linear Algebra Lecture 2 1.3.7 Matrix Matrix multiplication using Falk s
More 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 informationMath 1B03/1ZC3 - Tutorial 2. Jan. 21st/24th, 2014
Math 1B03/1ZC3 - Tutorial 2 Jan. 21st/24th, 2014 Tutorial Info: Website: http://ms.mcmaster.ca/ dedieula. Math Help Centre: Wednesdays 2:30-5:30pm. Email: dedieula@math.mcmaster.ca. Does the Commutative
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 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 informationsum of squared error.
IT 131 MATHEMATCS FOR SCIENCE LECTURE NOTE 6 LEAST SQUARES REGRESSION ANALYSIS and DETERMINANT OF A MATRIX Source: Larson, Edwards, Falvo (2009): Elementary Linear Algebra, Sixth Edition You will now look
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 informationCa Foscari University of Venice - Department of Management - A.A Luciano Battaia. December 14, 2017
Ca Foscari University of Venice - Department of Management - A.A.27-28 Mathematics Luciano Battaia December 4, 27 Brief summary for second partial - Sample Exercises Two variables functions Here and in
More informationProblems for M 10/26:
Math, Lesieutre Problem set # November 4, 25 Problems for M /26: 5 Is λ 2 an eigenvalue of 2? 8 Why or why not? 2 A 2I The determinant is, which means that A 2I has 6 a nullspace, and so there is an eigenvector
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 informationSection 4.5. Matrix Inverses
Section 4.5 Matrix Inverses The Definition of Inverse Recall: The multiplicative inverse (or reciprocal) of a nonzero number a is the number b such that ab = 1. We define the inverse of a matrix in almost
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 informationChapter 2. Square matrices
Chapter 2. Square matrices Lecture notes for MA1111 P. Karageorgis pete@maths.tcd.ie 1/18 Invertible matrices Definition 2.1 Invertible matrices An n n matrix A is said to be invertible, if there is a
More informationMATH 260 LINEAR ALGEBRA EXAM II Fall 2013 Instructions: The use of built-in functions of your calculator, such as det( ) or RREF, is prohibited.
MAH 60 LINEAR ALGEBRA EXAM II Fall 0 Instructions: he use of built-in functions of your calculator, such as det( ) or RREF, is prohibited ) For the matrix find: a) M and C b) M 4 and C 4 ) Evaluate the
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 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 informationChapter 4 - MATRIX ALGEBRA. ... a 2j... a 2n. a i1 a i2... a ij... a in
Chapter 4 - MATRIX ALGEBRA 4.1. Matrix Operations A a 11 a 12... a 1j... a 1n a 21. a 22.... a 2j... a 2n. a i1 a i2... a ij... a in... a m1 a m2... a mj... a mn The entry in the ith row and the jth column
More informationIntroduction to Matrices
POLS 704 Introduction to Matrices Introduction to Matrices. The Cast of Characters A matrix is a rectangular array (i.e., a table) of numbers. For example, 2 3 X 4 5 6 (4 3) 7 8 9 0 0 0 Thismatrix,with4rowsand3columns,isoforder
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 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 informationLinear Algebra (part 1) : Matrices and Systems of Linear Equations (by Evan Dummit, 2016, v. 2.02)
Linear Algebra (part ) : Matrices and Systems of Linear Equations (by Evan Dummit, 206, v 202) Contents 2 Matrices and Systems of Linear Equations 2 Systems of Linear Equations 2 Elimination, Matrix Formulation
More informationMatrices. Ellen Kulinsky
Matrices Ellen Kulinsky To learn the most (AKA become the smartest): Take notes. This is very important! I will sometimes tell you what to write down, but usually you will need to do it on your own. I
More informationEigenvalues and Eigenvectors
Contents Eigenvalues and Eigenvectors. Basic Concepts. Applications of Eigenvalues and Eigenvectors 8.3 Repeated Eigenvalues and Symmetric Matrices 3.4 Numerical Determination of Eigenvalues and Eigenvectors
More informationChapter 3. Determinants and Eigenvalues
Chapter 3. Determinants and Eigenvalues 3.1. Determinants With each square matrix we can associate a real number called the determinant of the matrix. Determinants have important applications to the theory
More informationLinear Algebra Practice Final
. Let (a) First, Linear Algebra Practice Final Summer 3 3 A = 5 3 3 rref([a ) = 5 so if we let x 5 = t, then x 4 = t, x 3 =, x = t, and x = t, so that t t x = t = t t whence ker A = span(,,,, ) and a basis
More information3 Matrix Algebra. 3.1 Operations on matrices
3 Matrix Algebra A matrix is a rectangular array of numbers; it is of size m n if it has m rows and n columns. A 1 n matrix is a row vector; an m 1 matrix is a column vector. For example: 1 5 3 5 3 5 8
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 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 informationChapter 9: Systems of Equations and Inequalities
Chapter 9: Systems of Equations and Inequalities 9. Systems of Equations Solve the system of equations below. By this we mean, find pair(s) of numbers (x, y) (if possible) that satisfy both equations.
More informationREVIEW FOR EXAM II. The exam covers sections , the part of 3.7 on Markov chains, and
REVIEW FOR EXAM II The exam covers sections 3.4 3.6, the part of 3.7 on Markov chains, and 4.1 4.3. 1. The LU factorization: An n n matrix A has an LU factorization if A = LU, where L is lower triangular
More informationThe Laplace Expansion Theorem: Computing the Determinants and Inverses of Matrices
The Laplace Expansion Theorem: Computing the Determinants and Inverses of Matrices David Eberly, Geometric Tools, Redmond WA 98052 https://www.geometrictools.com/ This work is licensed under the Creative
More informationIntroduction Growthequations Decay equations Forming differential equations Case studies Shifted equations Test INU0115/515 (MATHS 2)
GROWTH AND DECAY CALCULUS 12 INU0115/515 (MATHS 2) Dr Adrian Jannetta MIMA CMath FRAS Growth and Decay 1/ 24 Adrian Jannetta Introduction Some of the simplest systems that can be modelled by differential
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 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 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 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 informationJUST THE MATHS UNIT NUMBER 9.3. MATRICES 3 (Matrix inversion & simultaneous equations) A.J.Hobson
JUST THE MATHS UNIT NUMBER 93 MATRICES 3 (Matrix inversion & simultaneous equations) by AJHobson 931 Introduction 932 Matrix representation of simultaneous linear equations 933 The definition of a multiplicative
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 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 information1 Matrices and Systems of Linear Equations
Linear Algebra (part ) : Matrices and Systems of Linear Equations (by Evan Dummit, 207, v 260) Contents Matrices and Systems of Linear Equations Systems of Linear Equations Elimination, Matrix Formulation
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 informationDeterminant of a Matrix
13 March 2018 Goals We will define determinant of SQUARE matrices, inductively, using the definition of Minors and cofactors. We will see that determinant of triangular matrices is the product of its diagonal
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 informationProblems for M 11/2: A =
Math 30 Lesieutre Problem set # November 0 Problems for M /: 4 Let B be the basis given by b b Find the B-matrix for the transformation T : R R given by x Ax where 3 4 A (This just means the matrix for
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 informationJUST THE MATHS SLIDES NUMBER 9.3. MATRICES 3 (Matrix inversion & simultaneous equations) A.J.Hobson
JUST THE MATHS SLIDES NUMBER 93 MATRICES 3 (Matrix inversion & simultaneous equations) by AJHobson 93 Introduction 932 Matrix representation of simultaneous linear equations 933 The definition of a multiplicative
More informationDistance in the Plane
Distance in the Plane The absolute value function is defined as { x if x 0; and x = x if x < 0. If the number a is positive or zero, then a = a. If a is negative, then a is the number you d get by erasing
More informationMATH 1210 Assignment 4 Solutions 16R-T1
MATH 1210 Assignment 4 Solutions 16R-T1 Attempt all questions and show all your work. Due November 13, 2015. 1. Prove using mathematical induction that for any n 2, and collection of n m m matrices A 1,
More 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 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 informationSolution Set 7, Fall '12
Solution Set 7, 18.06 Fall '12 1. Do Problem 26 from 5.1. (It might take a while but when you see it, it's easy) Solution. Let n 3, and let A be an n n matrix whose i, j entry is i + j. To show that det
More informationAnnouncements Wednesday, October 04
Announcements Wednesday, October 04 Please fill out the mid-semester survey under Quizzes on Canvas. WeBWorK 1.8, 1.9 are due today at 11:59pm. The quiz on Friday covers 1.7, 1.8, and 1.9. My office is
More informationMatrix Inverses. November 19, 2014
Matrix Inverses November 9, 204 22 The Inverse of a Matrix Now that we have discussed how to multiply two matrices, we can finally have a proper discussion of what we mean by the expression A for a matrix
More informationMath Bootcamp An p-dimensional vector is p numbers put together. Written as. x 1 x =. x p
Math Bootcamp 2012 1 Review of matrix algebra 1.1 Vectors and rules of operations An p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the
More 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 informationSection 2.2: The Inverse of a Matrix
Section 22: The Inverse of a Matrix Recall that a linear equation ax b, where a and b are scalars and a 0, has the unique solution x a 1 b, where a 1 is the reciprocal of a From this result, it is natural
More information