12.3 Inverse Matrices

Size: px
Start display at page:

Download "12.3 Inverse Matrices"

Transcription

1 2.3 Inverse Matrices Two matrices A A are called inverses if AA I A A I where I denotes the identit matrix of the appropriate size. For example, the matrices A A If we think of the identit matrix I as being analogous to the number, then inverses are analogous to a reciprocal pair of numbers, such as 2/3 3/2. are inverses, since AA A A I I Here are some of the most important properties of inverse matrices:. The inverse of a matrix is unique. If A is a matrix, then there can be onl one possible inverse A for A. To underst wh, suppose that A is an inverse for A, suppose that B is also an inverse for A. Then You should be able to underst each step of this long equation. B BI BAA IA A so B must be the same matrix as A. 2. The inverse of the inverse is the original matrix. The relationship between A A is smmetric, so the inverse of A is the original matrix A. ( A ) A Non-square matrices whose product is the identit are sometimes called one-sided inverses. For square matrices, if the product AB is the identit matrix, then BA is alwas the identit as well, so A B are inverses of one another. 3. Onl square matrices can have inverses. Though it is possible for the product of two non-square matrices to be the identit matrix if we reverse the order of multiplication we no longer get the identit Onl for square matrices can both products ields the identit matrix.

2 INVERSE MATRICES 2 4. Not ever square matrix is invertible. Some square matrices do not have an inverse. For example, the zero matrix Sometimes invertible matrices are referred to as non-singular. certainl has no inverse, since the product of the zero matrix with an other matrix is alwas the zero matrix. there are also less obvious examples, such as the matrix which has no inverse. In general, a square matrix is called invertible if it has an inverse, singular if it does not. 5. The product of invertible matrices is invertible. If A B are invertible matrices, then the product AB is invertible as well, with (AB) B A Note that the order of the two matrices is reversed when we take the inverse. This order reversal is necessar to make the inverses cancel with AB when multiplied. In particular, ABB A AIA AA I B A AB B IB B B I Finding the Inverse There is a simple formula for the inverse of a 2 2 matrix: a b c d ad bc d c b a It is eas to check that this formula works. In particular, ad bc d c b a b a c d ad bc 0 ad bc 0 ad bc 0 0 EAMPLE Find the inverse of the matrix According to the formula above, the inverse is (3)(6) (2)(8) /2 Incidentall, the denominator ad bc in this formula is called the determinant of the 2 2 matrix. A 2 2 matrix whose determinant is zero is not invertible.

3 INVERSE MATRICES 3 Finding the inverse of a 3 3 or larger matrix is much more difficult. For example, the formula for the inverse of a 3 3 matrix is a b c d e f g h i aei a f h + b f g bdi + cdh ce g ei f h ch bi b f ce f g di ai c g cd a f dh e g b g ah ae bd This formula is so complicated that it is rarel worthwhile to compute the inverse of a 3 3 matrix b h, let alone larger sizes such as 4 4 or 5 5. Fortunatel, most modern calculators computer algebra sstems can compute the inverse of a matrix, several websites offer tools that let ou perform matrix computations online. Algebra of Inverse Matrices Inverse matrices let ou divide both sides of an equation b a matrix. For example, if we have a matrix equation of the form A B where is an unknown matrix, we can solve for b mutlipling both sides of the equation b A : A A A B Since A A I, the left side simplifies to just, so A B EAMPLE 2 Find a 2 2 matrix so that Here is the identit matrix, so is the same as I, which is just. which simplifies to so Multipling through b the inverse of gives (4)(2) (6)() 4 3 / /2 2 B the wa, since the order of matrix multiplication matters, one must be careful to multipl b A in the same wa on both sides of an equation. Given an equation Y

4 INVERSE MATRICES 4 we can either left-multipl b A on both sides A A Y or we can right-multipl b A on both sides A YA However, we cannot left-multipl b A on one side of the equation right-multipl b A on the other side. EAMPLE 3 Make sure to underst the difference between this example the previous one. Find a 2 2 matrix so that This time we must right-multipl both sides b the inverse of 4 6. This gives which simplifies to / , so / /2 2 Solving Linear Sstems with Inverses Recall that an n n linear sstem can be written in the form Ax b where A is the (square) matrix of coefficients, b is the vector of constant terms. If we know the inverse of A, we can solve this linear sstem b left-multipling both sides of the equation b A. This gives x A b Thus ou can solve a linear sstem b finding the inverse of the matrix of coefficients. This can be quite useful if ou have several linear sstems with the same coefficients but different constant terms. EAMPLE 4 Solve each of the following linear sstems: 3x x x + 2 5x x x + 3 5

5 INVERSE MATRICES 5 Each of these sstems is reall a vector equation: x x Thus the solutions are respectivel x x so the solutions are x (3)(3) (2)(5) x 3 2 x x x EERCISES 4 Find the inverse of the given matrix, or state that the matrix is not invertible Find scalars a, b, c so that the given pair of matrices are inverses , a b 2 3 c , 3 8 a 7 b 6 c 2 7. Suppose that A, B, C are invertible n n matrices. (a) Is the product ABC invertible? If so, what is its inverse? (b) Is AB C invertible? If so, what is its inverse? 8 Find a 2 2 matrix that satisfies the given equation Use inverse matrices to find the solution to the each of the following linear sstems: 4x x x x x x + 3

MAT188H1S Lec0101 Burbulla

MAT188H1S Lec0101 Burbulla Winter 206 Linear Transformations A linear transformation T : R m R n is a function that takes vectors in R m to vectors in R n such that and T (u + v) T (u) + T (v) T (k v) k T (v), for all vectors u

More information

Math 313 Lecture #10 2.2: The Inverse of a Matrix

Math 313 Lecture #10 2.2: The Inverse of a Matrix Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is

More information

Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M 1 such that

Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M 1 such that 0. Inverse Matrix Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M such that M M = I = M M. Inverse of a 2 2 Matrix Let M and N be the matrices: a b d b M =, N = c

More information

Matrices: 2.3 The Inverse of Matrices

Matrices: 2.3 The Inverse of Matrices September 4 Goals Define inverse of a matrix. Point out that not every matrix A has an inverse. Discuss uniqueness of inverse of a matrix A. Discuss methods of computing inverses, particularly by row operations.

More information

Math 2331 Linear Algebra

Math 2331 Linear Algebra 2.2 The Inverse of a Matrix Math 2331 Linear Algebra 2.2 The Inverse of a Matrix Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math2331 Jiwen He, University

More information

Introduction to Matrices for Engineers

Introduction to Matrices for Engineers Introduction to Matrices for Engineers C.T.J. Dodson, School of Mathematics, Manchester Universit 1 What is a Matrix? A matrix is a rectangular arra of elements, usuall numbers, e.g. 1 0-8 4 0-1 1 0 11

More information

B such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix

B such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix Matrix inverses Recall... Definition A square matrix A is invertible (or nonsingular) if matrix B such that AB = and BA =. (We say B is an inverse of A.) Remark Not all square matrices are invertible.

More information

Inverses and powers: Rules of Matrix Arithmetic

Inverses and powers: Rules of Matrix Arithmetic Contents 1 Inverses and powers: Rules of Matrix Arithmetic 1.1 What about division of matrices? 1.2 Properties of the Inverse of a Matrix 1.2.1 Theorem (Uniqueness of Inverse) 1.2.2 Inverse Test 1.2.3

More information

Lecture 2 Matrix Operations

Lecture 2 Matrix Operations Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrix-vector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or

More information

Matrices, transposes, and inverses

Matrices, transposes, and inverses Matrices, transposes, and inverses Math 40, Introduction to Linear Algebra Wednesday, February, 202 Matrix-vector multiplication: two views st perspective: A x is linear combination of columns of A 2 4

More information

MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.

MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted

More information

NON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that

NON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that NON SINGULAR MATRICES DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that AB = I n = BA. Any matrix B with the above property is called

More information

Homework: 2.1 (page 56): 7, 9, 13, 15, 17, 25, 27, 35, 37, 41, 46, 49, 67

Homework: 2.1 (page 56): 7, 9, 13, 15, 17, 25, 27, 35, 37, 41, 46, 49, 67 Chapter Matrices Operations with Matrices Homework: (page 56):, 9, 3, 5,, 5,, 35, 3, 4, 46, 49, 6 Main points in this section: We define a few concept regarding matrices This would include addition of

More information

Determinants. Dr. Doreen De Leon Math 152, Fall 2015

Determinants. Dr. Doreen De Leon Math 152, Fall 2015 Determinants Dr. Doreen De Leon Math 52, Fall 205 Determinant of a Matrix Elementary Matrices We will first discuss matrices that can be used to produce an elementary row operation on a given matrix A.

More information

Inverse Relations and Functions

Inverse Relations and Functions 11.1 Inverse Relations and Functions 11.1 OBJECTIVES 1. Find the inverse of a relation 2. Graph a relation and its inverse 3. Find the inverse of a function 4. Graph a function and its inverse 5. Identif

More information

Linear Dependence Tests

Linear Dependence Tests Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks

More information

9 Matrices, determinants, inverse matrix, Cramer s Rule

9 Matrices, determinants, inverse matrix, Cramer s Rule AAC - Business Mathematics I Lecture #9, December 15, 2007 Katarína Kálovcová 9 Matrices, determinants, inverse matrix, Cramer s Rule Basic properties of matrices: Example: Addition properties: Associative:

More information

UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

More information

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

More information

The Inverse of a Matrix

The Inverse of a Matrix The Inverse of a Matrix 7.4 Introduction In number arithmetic every number a ( 0) has a reciprocal b written as a or such that a ba = ab =. Some, but not all, square matrices have inverses. If a square

More information

MathQuest: Linear Algebra. 1. Which of the following matrices does not have an inverse?

MathQuest: Linear Algebra. 1. Which of the following matrices does not have an inverse? MathQuest: Linear Algebra Matrix Inverses 1. Which of the following matrices does not have an inverse? 1 2 (a) 3 4 2 2 (b) 4 4 1 (c) 3 4 (d) 2 (e) More than one of the above do not have inverses. (f) All

More information

( % . This matrix consists of $ 4 5 " 5' the coefficients of the variables as they appear in the original system. The augmented 3 " 2 2 # 2 " 3 4&

( % . This matrix consists of $ 4 5  5' the coefficients of the variables as they appear in the original system. The augmented 3  2 2 # 2  3 4& Matrices define matrix We will use matrices to help us solve systems of equations. A matrix is a rectangular array of numbers enclosed in parentheses or brackets. In linear algebra, matrices are important

More information

SECTION 8.3: THE INVERSE OF A SQUARE MATRIX

SECTION 8.3: THE INVERSE OF A SQUARE MATRIX (Section 8.3: The Inverse of a Square Matrix) 8.47 SECTION 8.3: THE INVERSE OF A SQUARE MATRIX PART A: (REVIEW) THE INVERSE OF A REAL NUMBER If a is a nonzero real number, then aa 1 = a 1 a = 1. a 1, or

More information

1.5 Elementary Matrices and a Method for Finding the Inverse

1.5 Elementary Matrices and a Method for Finding the Inverse .5 Elementary Matrices and a Method for Finding the Inverse Definition A n n matrix is called an elementary matrix if it can be obtained from I n by performing a single elementary row operation Reminder:

More information

LECTURE 1 I. Inverse matrices We return now to the problem of solving linear equations. Recall that we are trying to find x such that IA = A

LECTURE 1 I. Inverse matrices We return now to the problem of solving linear equations. Recall that we are trying to find x such that IA = A LECTURE I. Inverse matrices We return now to the problem of solving linear equations. Recall that we are trying to find such that A = y. Recall: there is a matri I such that for all R n. It follows that

More information

2.1: MATRIX OPERATIONS

2.1: MATRIX OPERATIONS .: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and

More information

MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix.

MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. Matrices Definition. An m-by-n matrix is a rectangular array of numbers that has m rows and n columns: a 11

More information

The Inverse of a Square Matrix

The Inverse of a Square Matrix These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for in-class presentation

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 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

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

More information

Solving Linear Systems, Continued and The Inverse of a Matrix

Solving Linear Systems, Continued and The Inverse of a Matrix , Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing

More information

Chapter 8. Matrices II: inverses. 8.1 What is an inverse?

Chapter 8. Matrices II: inverses. 8.1 What is an inverse? Chapter 8 Matrices II: inverses We have learnt how to add subtract and multiply matrices but we have not defined division. The reason is that in general it cannot always be defined. In this chapter, we

More information

1 Introduction to Matrices

1 Introduction to Matrices 1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns

More information

Inverses. Stephen Boyd. EE103 Stanford University. October 27, 2015

Inverses. Stephen Boyd. EE103 Stanford University. October 27, 2015 Inverses Stephen Boyd EE103 Stanford University October 27, 2015 Outline Left and right inverses Inverse Solving linear equations Examples Pseudo-inverse Left and right inverses 2 Left inverses a number

More information

Lecture 10: Invertible matrices. Finding the inverse of a matrix

Lecture 10: Invertible matrices. Finding the inverse of a matrix Lecture 10: Invertible matrices. Finding the inverse of a matrix Danny W. Crytser April 11, 2014 Today s lecture Today we will Today s lecture Today we will 1 Single out a class of especially nice matrices

More information

x y The matrix form, the vector form, and the augmented matrix form, respectively, for the system of equations are

x y The matrix form, the vector form, and the augmented matrix form, respectively, for the system of equations are Solving Sstems of Linear Equations in Matri Form with rref Learning Goals Determine the solution of a sstem of equations from the augmented matri Determine the reduced row echelon form of the augmented

More information

4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns

4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows

More information

Lecture 6. Inverse of Matrix

Lecture 6. Inverse of Matrix Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,

More information

Matrix Algebra and Applications

Matrix Algebra and Applications Matrix Algebra and Applications Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 1 / 49 EC2040 Topic 2 - Matrices and Matrix Algebra Reading 1 Chapters

More information

Simplification of Rational Expressions and Functions

Simplification of Rational Expressions and Functions 7.1 Simplification of Rational Epressions and Functions 7.1 OBJECTIVES 1. Simplif a rational epression 2. Identif a rational function 3. Simplif a rational function 4. Graph a rational function Our work

More information

Lecture 21: The Inverse of a Matrix

Lecture 21: The Inverse of a Matrix Lecture 21: The Inverse of a Matrix Winfried Just, Ohio University October 16, 2015 Review: Our chemical reaction system Recall our chemical reaction system A + 2B 2C A + B D A + 2C 2D B + D 2C If we write

More information

8.7 Systems of Non-Linear Equations and Inequalities

8.7 Systems of Non-Linear Equations and Inequalities 8.7 Sstems of Non-Linear Equations and Inequalities 67 8.7 Sstems of Non-Linear Equations and Inequalities In this section, we stud sstems of non-linear equations and inequalities. Unlike the sstems of

More information

7.3 Solving Systems by Elimination

7.3 Solving Systems by Elimination 7. Solving Sstems b Elimination In the last section we saw the Substitution Method. It turns out there is another method for solving a sstem of linear equations that is also ver good. First, we will need

More information

(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.

(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product

More information

Zero and Negative Exponents and Scientific Notation. a a n a m n. Now, suppose that we allow m to equal n. We then have. a am m a 0 (1) a m

Zero and Negative Exponents and Scientific Notation. a a n a m n. Now, suppose that we allow m to equal n. We then have. a am m a 0 (1) a m 0. E a m p l e 666SECTION 0. OBJECTIVES. Define the zero eponent. Simplif epressions with negative eponents. Write a number in scientific notation. Solve an application of scientific notation We must have

More information

Linear Algebraic Equations, SVD, and the Pseudo-Inverse

Linear Algebraic Equations, SVD, and the Pseudo-Inverse Linear Algebraic Equations, SVD, and the Pseudo-Inverse Philip N. Sabes October, 21 1 A Little Background 1.1 Singular values and matrix inversion For non-smmetric matrices, the eigenvalues and singular

More information

Graphing Quadratic Equations

Graphing Quadratic Equations .4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph first-degree equations. Similar methods will allow ou to graph quadratic equations

More information

Row Operations and Inverse Matrices on the TI-83

Row Operations and Inverse Matrices on the TI-83 Row Operations and Inverse Matrices on the TI-83 I. Elementary Row Operations 2 8 A. Let A =. 2 7 B. To interchange rows and 2 of matrix A: MATRIX MATH C:rowSwap( MATRIX NAMES :[A],, 2 ) ENTER. 2 7 The

More information

Graphing Linear Inequalities in Two Variables

Graphing Linear Inequalities in Two Variables 5.4 Graphing Linear Inequalities in Two Variables 5.4 OBJECTIVES 1. Graph linear inequalities in two variables 2. Graph a region defined b linear inequalities What does the solution set look like when

More information

Partitioned Matrices and the Schur Complement

Partitioned Matrices and the Schur Complement P Partitioned Matrices and the Schur Complement P 1 Appendix P: PARTITIONED MATRICES AND THE SCHUR COMPLEMENT TABLE OF CONTENTS Page P1 Partitioned Matrix P 3 P2 Schur Complements P 3 P3 Block Diagonalization

More information

Lecture 23: The Inverse of a Matrix

Lecture 23: The Inverse of a Matrix Lecture 23: The Inverse of a Matrix Winfried Just, Ohio University March 9, 2016 The definition of the matrix inverse Let A be an n n square matrix. The inverse of A is an n n matrix A 1 such that A 1

More information

Typical Linear Equation Set and Corresponding Matrices

Typical Linear Equation Set and Corresponding Matrices EWE: Engineering With Excel Larsen Page 1 4. Matrix Operations in Excel. Matrix Manipulations: Vectors, Matrices, and Arrays. How Excel Handles Matrix Math. Basic Matrix Operations. Solving Systems of

More information

Using row reduction to calculate the inverse and the determinant of a square matrix

Using row reduction to calculate the inverse and the determinant of a square matrix Using row reduction to calculate the inverse and the determinant of a square matrix Notes for MATH 0290 Honors by Prof. Anna Vainchtein 1 Inverse of a square matrix An n n square matrix A is called invertible

More information

Find the Relationship: An Exercise in Graphing Analysis

Find the Relationship: An Exercise in Graphing Analysis Find the Relationship: An Eercise in Graphing Analsis Computer 5 In several laborator investigations ou do this ear, a primar purpose will be to find the mathematical relationship between two variables.

More information

Chapter 7. Matrices. Definition. An m n matrix is an array of numbers set out in m rows and n columns. Examples. ( 1 1 5 2 0 6

Chapter 7. Matrices. Definition. An m n matrix is an array of numbers set out in m rows and n columns. Examples. ( 1 1 5 2 0 6 Chapter 7 Matrices Definition An m n matrix is an array of numbers set out in m rows and n columns Examples (i ( 1 1 5 2 0 6 has 2 rows and 3 columns and so it is a 2 3 matrix (ii 1 0 7 1 2 3 3 1 is a

More information

Mathematics Notes for Class 12 chapter 3. Matrices

Mathematics Notes for Class 12 chapter 3. Matrices 1 P a g e Mathematics Notes for Class 12 chapter 3. Matrices A matrix is a rectangular arrangement of numbers (real or complex) which may be represented as matrix is enclosed by [ ] or ( ) or Compact form

More information

Helpsheet. Giblin Eunson Library MATRIX ALGEBRA. library.unimelb.edu.au/libraries/bee. Use this sheet to help you:

Helpsheet. Giblin Eunson Library MATRIX ALGEBRA. library.unimelb.edu.au/libraries/bee. Use this sheet to help you: Helpsheet Giblin Eunson Library ATRIX ALGEBRA Use this sheet to help you: Understand the basic concepts and definitions of matrix algebra Express a set of linear equations in matrix notation Evaluate determinants

More information

The Quadratic Function

The Quadratic Function 0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral

More information

Basic Properties of Rings

Basic Properties of Rings Basic Properties of Rings A ring is an algebraic structure with an addition operation and a multiplication operation. These operations are required to satisfy many (but not all!) familiar properties. Some

More information

Matrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.

Matrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such that

More information

Lecture Notes: Matrix Inverse. 1 Inverse Definition

Lecture Notes: Matrix Inverse. 1 Inverse Definition Lecture Notes: Matrix Inverse Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk Inverse Definition We use I to represent identity matrices,

More information

5.2 Inverse Functions

5.2 Inverse Functions 78 Further Topics in Functions. Inverse Functions Thinking of a function as a process like we did in Section., in this section we seek another function which might reverse that process. As in real life,

More information

The Graph of a Linear Equation

The Graph of a Linear Equation 4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that

More information

The Rectangular Coordinate System

The Rectangular Coordinate System The Mathematics Competenc Test The Rectangular Coordinate Sstem When we write down a formula for some quantit,, in terms of another quantit,, we are epressing a relationship between the two quantities.

More information

Solving Systems of Linear Equations. Substitution

Solving Systems of Linear Equations. Substitution Solving Systems of Linear Equations There are two basic methods we will use to solve systems of linear equations: Substitution Elimination We will describe each for a system of two equations in two unknowns,

More information

L1-2. Special Matrix Operations: Permutations, Transpose, Inverse, Augmentation 12 Aug 2014

L1-2. Special Matrix Operations: Permutations, Transpose, Inverse, Augmentation 12 Aug 2014 L1-2. Special Matrix Operations: Permutations, Transpose, Inverse, Augmentation 12 Aug 2014 Unfortunately, no one can be told what the Matrix is. You have to see it for yourself. -- Morpheus Primary concepts:

More information

Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses

Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 1 of 57 Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses Peter J. Hammond email: p.j.hammond@warwick.ac.uk Autumn 2012,

More information

Introduction to Matrix Algebra I

Introduction to Matrix Algebra I Appendix A Introduction to Matrix Algebra I Today we will begin the course with a discussion of matrix algebra. Why are we studying this? We will use matrix algebra to derive the linear regression model

More information

EC9A0: Pre-sessional Advanced Mathematics Course

EC9A0: Pre-sessional Advanced Mathematics Course University of Warwick, EC9A0: Pre-sessional Advanced Mathematics Course Peter J. Hammond & Pablo F. Beker 1 of 55 EC9A0: Pre-sessional Advanced Mathematics Course Slides 1: Matrix Algebra Peter J. Hammond

More information

Chapter 7: Eigenvalues and Eigenvectors 16

Chapter 7: Eigenvalues and Eigenvectors 16 Chapter 7: Eigenvalues and Eigenvectors 6 SECION G Sketching Conics B the end of this section ou will be able to recognise equations of different tpes of conics complete the square and use this to sketch

More information

To Be or Not To Be a Linear Equation: That Is the Question

To Be or Not To Be a Linear Equation: That Is the Question To Be or Not To Be a Linear Equation: That Is the Question Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form A + B C where A and B are not

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Chapter Section.1 Quadratic Functions Polnomial and Rational Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Course Number Instructor Date Important

More information

Basics Inversion and related concepts Random vectors Matrix calculus. Matrix algebra. Patrick Breheny. January 20

Basics Inversion and related concepts Random vectors Matrix calculus. Matrix algebra. Patrick Breheny. January 20 Matrix algebra January 20 Introduction Basics The mathematics of multiple regression revolves around ordering and keeping track of large arrays of numbers and solving systems of equations The mathematical

More information

MATH 105: Finite Mathematics 2-6: The Inverse of a Matrix

MATH 105: Finite Mathematics 2-6: The Inverse of a Matrix MATH 05: Finite Mathematics 2-6: The Inverse of a Matrix Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006 Outline Solving a Matrix Equation 2 The Inverse of a Matrix 3 Solving Systems of

More information

A matrix over a field F is a rectangular array of elements from F. The symbol

A matrix over a field F is a rectangular array of elements from F. The symbol Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted

More information

Math 1400/1650 (Cherry): Quadratic Functions

Math 1400/1650 (Cherry): Quadratic Functions Math 100/1650 (Cherr): Quadratic Functions A quadratic function is a function Q() of the form Q() = a + b + c with a 0. For eample, Q() = 3 7 + 5 is a quadratic function, and here a = 3, b = 7 and c =

More information

Undergraduate Matrix Theory. Linear Algebra

Undergraduate Matrix Theory. Linear Algebra Undergraduate Matrix Theory and Linear Algebra a 11 a 12 a 1n a 21 a 22 a 2n a m1 a m2 a mn John S Alin Linfield College Colin L Starr Willamette University December 15, 2015 ii Contents 1 SYSTEMS OF LINEAR

More information

Calculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants

Calculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants Calculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants Hartmut Führ fuehr@matha.rwth-aachen.de Lehrstuhl A für Mathematik, RWTH Aachen October 30, 2008 Overview

More information

15.062 Data Mining: Algorithms and Applications Matrix Math Review

15.062 Data Mining: Algorithms and Applications Matrix Math Review .6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop

More information

EIGENVALUES AND EIGENVECTORS

EIGENVALUES AND EIGENVECTORS Chapter 6 EIGENVALUES AND EIGENVECTORS 61 Motivation We motivate the chapter on eigenvalues b discussing the equation ax + hx + b = c, where not all of a, h, b are zero The expression ax + hx + b is called

More information

Question 2: How do you solve a matrix equation using the matrix inverse?

Question 2: How do you solve a matrix equation using the matrix inverse? Question : How do you solve a matrix equation using the matrix inverse? In the previous question, we wrote systems of equations as a matrix equation AX B. In this format, the matrix A contains the coefficients

More information

Chapter Adequacy of Solutions

Chapter Adequacy of Solutions Chapter 04.09 dequac of Solutions fter reading this chapter, ou should be able to:. know the difference between ill-conditioned and well-conditioned sstems of equations,. define and find the norm of a

More information

Graphing Quadratic Functions

Graphing Quadratic Functions A. THE STANDARD PARABOLA Graphing Quadratic Functions The graph of a quadratic function is called a parabola. The most basic graph is of the function =, as shown in Figure, and it is to this graph which

More information

Graphing Linear Equations

Graphing Linear Equations 6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are

More information

Chapter 4: Binary Operations and Relations

Chapter 4: Binary Operations and Relations c Dr Oksana Shatalov, Fall 2014 1 Chapter 4: Binary Operations and Relations 4.1: Binary Operations DEFINITION 1. A binary operation on a nonempty set A is a function from A A to A. Addition, subtraction,

More information

Identifying second degree equations

Identifying second degree equations Chapter 7 Identifing second degree equations 7.1 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +

More information

STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS

STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS The intermediate algebra skills illustrated here will be used extensively and regularly throughout the semester Thus, mastering these skills is an

More information

Use order of operations to simplify. Show all steps in the space provided below each problem. INTEGER OPERATIONS

Use order of operations to simplify. Show all steps in the space provided below each problem. INTEGER OPERATIONS ORDER OF OPERATIONS In the following order: 1) Work inside the grouping smbols such as parenthesis and brackets. ) Evaluate the powers. 3) Do the multiplication and/or division in order from left to right.

More information

C1: Coordinate geometry of straight lines

C1: Coordinate geometry of straight lines B_Chap0_08-05.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the

More information

Divisibility of the Determinant of a Class of Matrices with Integer Entries

Divisibility of the Determinant of a Class of Matrices with Integer Entries Divisibility of the Determinant of a Class of Matrices with Integer Entries Sujan Pant and Dennis I. Merino August 8, 010 Abstract Let x i = n j=1 a ij10 n j, with i = 1,..., n, and with each a ij Z be

More information

Cofactor Expansion: Cramer s Rule

Cofactor Expansion: Cramer s Rule Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating

More information

A general system of m equations in n unknowns. a 11 x 1 + a 12 x a 1n x n = a 21 x 1 + a 22 x a 2n x n =

A general system of m equations in n unknowns. a 11 x 1 + a 12 x a 1n x n = a 21 x 1 + a 22 x a 2n x n = A general system of m equations in n unknowns MathsTrack (NOTE Feb 2013: This is the old version of MathsTrack. New books will be created during 2013 and 2014) a 11 x 1 + a 12 x 2 + + a 1n x n a 21 x 1

More information

Q (x 1, y 1 ) m = y 1 y 0

Q (x 1, y 1 ) m = y 1 y 0 . Linear Functions We now begin the stud of families of functions. Our first famil, linear functions, are old friends as we shall soon see. Recall from Geometr that two distinct points in the plane determine

More information

Chapter 6: Radical Functions and Rational Exponents

Chapter 6: Radical Functions and Rational Exponents Algebra B: Chapter 6 Notes 1 Chapter 6: Radical Functions and Rational Eponents Concept Bte (Review): Properties of Eponents Recall from Algebra 1, the Properties (Rules) of Eponents. Propert of Eponents:

More information

ALGEBRA 2 SCOPE AND SEQUENCE. Writing Quadratic Equations Quadratic Regression

ALGEBRA 2 SCOPE AND SEQUENCE. Writing Quadratic Equations Quadratic Regression ALGEBRA 2 SCOPE AND SEQUENCE UNIT 1 2 3 4 5 6 DATES & NO. OF DAYS 8/22-9/16 19 days 9/19-9/30 9 days 10/3-10/14 9 days 10/20-10/25 4 days 10/27-12/2 17 days 12/5-12/16 10 days UNIT NAME Foundations for

More information

We know a formula for and some properties of the determinant. Now we see how the determinant can be used.

We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Cramer s rule, inverse matrix, and volume We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Formula for A We know: a b d b =. c d ad bc c a Can we

More information

Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

More information

Lesson 8.3 Exercises, pages

Lesson 8.3 Exercises, pages Lesson 8. Eercises, pages 57 5 A. For each function, write the equation of the corresponding reciprocal function. a) = 5 - b) = 5 c) = - d) =. Sketch broken lines to represent the vertical and horizontal

More information

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b

More information