Lecture Notes: Matrix Inverse. 1 Inverse Definition

Size: px
Start display at page:

Download "Lecture Notes: Matrix Inverse. 1 Inverse Definition"

Transcription

1 Lecture Notes: Matrix Inverse Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong Inverse Definition We use I to represent identity matrices, namely, diagonal matrices where all the elements on the main diagonal are. Definition. Let A and B be n n matrices. If AB = I, then we say that B is the inverse of A, denoted as A. For example, let A = You can verify that AB = 4,B = -4-4 /2. Hence, B = A. We state the next non-trivial but well-known result without proof: Lemma. Let A and B be n n matrices. If AB = I, then BA = I. In other words, if B is A, then A = B. The lemma has an important corollary: Corollary. Let A be an n n matrix. If A exists, then it is unique. Proof. Suppose that there were two different matrices B and C satisfying AB = I and AC = I. Then, by Lemma, we have: which is a contradiction. BA = I BAC = IC B(AC) = C BI = C B = C

2 It is important to note that some square matrices have no inverse. For example, has no inverse (you are encouraged to make an attempt to find it, and see where you will get stuck). The next lemma gives the precise condition for the existence of inverses: Lemma 2. An n n matrix A has an inverse if and only if det(a). Proof. If-Direction. If det(a), then the linear system Ax = b has a unique solution for any b. Denote by b i the i-th column ( i n) of the n n identity matrix I, and x i the solution of the system Ax i = b i. Then, we obtain A by placing x i as the i-th column of A, for each i [,n. Only-If Direction. If A exists, then given any linear system Ax = b, we have A Ax = A b, which means x = A b. This means that Ax = b has at least one solution. Furthermore, by the uniqueness of A, we know that Ax = b has a unique solution. It thus follows that the rank of A is n, namely, det(a). An n n matrix A is said to be singular if it does not have an inverse; non-singular if it does. 2 Properties of Inverse Lemma 3. Let A,B be n n non-singular matrices. Then, (AB) = B A. Proof. Hence, (AB) = B A. (AB)(B A ) = A(BB )A = AIA = AA = I. Lemma 4. Let A be an n n non-singular matrix. Then, (A T ) = (A ) T. Proof. It suffices to prove that A T (A ) T = I, which is equivalent to proving (A T (A ) T ) T = I T = I. This is true because (A T (A ) T ) T = A A = I. Lemma 5. Let A be an n n non-singular matrix. Then, det(a ) = /det(a). Proof. det(a) det(a ) = det(aa ) = det(i) =. The lemma thus follows. Lemma 6. Let A,B and C be n n matrices. If A is non-singular and AB = AC, then B = C. 2

3 Proof. From AB = AC, we have A AB = A AC, which gives B = C. Lemma 7. Let A and B be n n matrices such that AB = (where is the zero matrix). Then:. If A is non-singular, then B =. 2. If B is non-singular, then A =. 3. (Corollary) If neither A nor B is, then A and B are both singular. Proof. Statement. If A is non-singular, then we have A AB = A, which gives B =. Statement 2. If B is non-singular, then we have ABB = B, which gives A =. 3 Inverse Computation: Gauss-Jordan Elimination We will use an example to illustrate how to compute the inverse of a matrix A. Consider that A = 4 Suppose that we do not know what is A ; hence, we assume: x x 2 x 3 A = x 2 x 22 x 23 x 3 x 32 x 33 Remember that we want 4 x x 2 x 3 x 2 x 22 x 23 x 3 x 32 x 33 = This is essentially to solve three linear systems: x x 2 x 3 x 2 x 22 x 32 x 3 x 23 x 33 = = = () (2) (3) Now we can focus on solving these systems respectively using Gauss Elimination. For example, to solve the linear system (), we look at the augmented matrix: (4) 3

4 Usually, we would start back substitution from here, but now we take a different approach. In particular, we will show that (since the system has a unique solution) it is possible to get rid of back substitution, but instead, continue to use elementary row operations to make the left side of the vertical bar an identity matrix. Then, the solution of the system will present itself. Specifically: (4) 4 /2-4 /2 It is thus clear that x = 4,x 2 = /2,x 3 =. The above method is an extension of Gauss elimination, and is referred to as Gauss-Jordan elimination. Now you may proceed to solve (2) and (3) in the same way. You will then realize that the operations done to the left of the vertical line are always the same. Motivated by this, we can solve all three systems ()-(3) together in one go, as illustrated below: / /2 What is now on the right side of the bar is exactly A. It is important to observe that the above process has in fact embedded the Gauss-Jordan elimination for solving all three linear systems ()-(3). 4 Inverse Formula It is possible to give a general formula for the inverse of an n n non-singular matrix A. As before, given i,j [,n, we denote by M ij the (n ) (n ) matrix obtained from A after discarding its i-th row and j-th column. Also, define: Then we have: C ij = ( ) i+j det(m ij ). Lemma 8. A = det(a) C C 2... C n C 2 C C n C n C 2n... C nn. 4

5 We skip a proof of the lemma, but illustrate it with an example. Example. Consider once again A = 4, We have: det(a) = 2. Also: M = M 2 = M 3 = M 2 = M 22 = M 23 = M 3 = M 32 = M 33 = [ 4-2 [ 4 [ -2 [ 2-2 [ [ 2-2 [ 2 4 [ 4 [ 2, and thus C = 8, and C 2 =, and C 3 = 2, and C 2 = 2, and C 22 =, and C 23 =, and C 3 = 8, and C 32 =, and C 33 = 2 Therefore, by Lemma 8, we have: A = det(a) = 2 C C 2 C 3 C 2 C 22 C 32 C 3 C 23 C =. 4 4 /2. 5

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

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

( % . 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

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

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

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 11. Shuanglin Shao. October 2nd and 7th, 2013

Lecture 11. Shuanglin Shao. October 2nd and 7th, 2013 Lecture 11 Shuanglin Shao October 2nd and 7th, 2013 Matrix determinants: addition. Determinants: multiplication. Adjoint of a matrix. Cramer s rule to solve a linear system. Recall that from the previous

More information

Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.

Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i. Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(

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

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

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

Diagonal, Symmetric and Triangular Matrices

Diagonal, Symmetric and Triangular Matrices Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by

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

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

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

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

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

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

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

Topic 1: Matrices and Systems of Linear Equations.

Topic 1: Matrices and Systems of Linear Equations. Topic 1: Matrices and Systems of Linear Equations Let us start with a review of some linear algebra concepts we have already learned, such as matrices, determinants, etc Also, we shall review the method

More information

1 Determinants. Definition 1

1 Determinants. Definition 1 Determinants The determinant of a square matrix is a value in R assigned to the matrix, it characterizes matrices which are invertible (det 0) and is related to the volume of a parallelpiped described

More information

Solution. Area(OABC) = Area(OAB) + Area(OBC) = 1 2 det( [ 5 2 1 2. Question 2. Let A = (a) Calculate the nullspace of the matrix A.

Solution. Area(OABC) = Area(OAB) + Area(OBC) = 1 2 det( [ 5 2 1 2. Question 2. Let A = (a) Calculate the nullspace of the matrix A. Solutions to Math 30 Take-home prelim Question. Find the area of the quadrilateral OABC on the figure below, coordinates given in brackets. [See pp. 60 63 of the book.] y C(, 4) B(, ) A(5, ) O x Area(OABC)

More information

Matrix Inverse and Determinants

Matrix Inverse and Determinants DM554 Linear and Integer Programming Lecture 5 and Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1 2 3 4 and Cramer s rule 2 Outline 1 2 3 4 and

More information

1. For each of the following matrices, determine whether it is in row echelon form, reduced row echelon form, or neither.

1. For each of the following matrices, determine whether it is in row echelon form, reduced row echelon form, or neither. Math Exam - Practice Problem Solutions. For each of the following matrices, determine whether it is in row echelon form, reduced row echelon form, or neither. (a) 5 (c) Since each row has a leading that

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

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

T ( a i x i ) = a i T (x i ).

T ( a i x i ) = a i T (x i ). Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)

More information

Math 315: Linear Algebra Solutions to Midterm Exam I

Math 315: Linear Algebra Solutions to Midterm Exam I Math 35: Linear Algebra s to Midterm Exam I # Consider the following two systems of linear equations (I) ax + by = k cx + dy = l (II) ax + by = 0 cx + dy = 0 (a) Prove: If x = x, y = y and x = x 2, y =

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

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

Notes on Determinant

Notes on Determinant ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

More information

Lecture 4: Partitioned Matrices and Determinants

Lecture 4: Partitioned Matrices and Determinants Lecture 4: Partitioned Matrices and Determinants 1 Elementary row operations Recall the elementary operations on the rows of a matrix, equivalent to premultiplying by an elementary matrix E: (1) multiplying

More information

INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL

INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL SOLUTIONS OF THEORETICAL EXERCISES selected from INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL Eighth Edition, Prentice Hall, 2005. Dr. Grigore CĂLUGĂREANU Department of Mathematics

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

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

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

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

DETERMINANTS. b 2. x 2

DETERMINANTS. b 2. x 2 DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in

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

SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89. by Joseph Collison

SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89. by Joseph Collison SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89 by Joseph Collison Copyright 2000 by Joseph Collison All rights reserved Reproduction or translation of any part of this work beyond that permitted by Sections

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

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

1 Eigenvalues and Eigenvectors

1 Eigenvalues and Eigenvectors Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x

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

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

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.

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. Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

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

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

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

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

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

1 Gaussian Elimination

1 Gaussian Elimination Contents 1 Gaussian Elimination 1.1 Elementary Row Operations 1.2 Some matrices whose associated system of equations are easy to solve 1.3 Gaussian Elimination 1.4 Gauss-Jordan reduction and the Reduced

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

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

Chapter 1 - Matrices & Determinants

Chapter 1 - Matrices & Determinants Chapter 1 - Matrices & Determinants Arthur Cayley (August 16, 1821 - January 26, 1895) was a British Mathematician and Founder of the Modern British School of Pure Mathematics. As a child, Cayley enjoyed

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

8 Square matrices continued: Determinants

8 Square matrices continued: Determinants 8 Square matrices continued: Determinants 8. Introduction Determinants give us important information about square matrices, and, as we ll soon see, are essential for the computation of eigenvalues. You

More information

= [a ij ] 2 3. Square matrix A square matrix is one that has equal number of rows and columns, that is n = m. Some examples of square matrices are

= [a ij ] 2 3. Square matrix A square matrix is one that has equal number of rows and columns, that is n = m. Some examples of square matrices are This document deals with the fundamentals of matrix algebra and is adapted from B.C. Kuo, Linear Networks and Systems, McGraw Hill, 1967. It is presented here for educational purposes. 1 Introduction In

More information

MATH 240 Fall, Chapter 1: Linear Equations and Matrices

MATH 240 Fall, Chapter 1: Linear Equations and Matrices MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS

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

In this leaflet we explain what is meant by an inverse matrix and how it is calculated.

In this leaflet we explain what is meant by an inverse matrix and how it is calculated. 5.5 Introduction The inverse of a matrix In this leaflet we explain what is meant by an inverse matrix and how it is calculated. 1. The inverse of a matrix The inverse of a square n n matrix A, is another

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

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

MATH 2030: SYSTEMS OF LINEAR EQUATIONS. ax + by + cz = d. )z = e. while these equations are not linear: xy z = 2, x x = 0,

MATH 2030: SYSTEMS OF LINEAR EQUATIONS. ax + by + cz = d. )z = e. while these equations are not linear: xy z = 2, x x = 0, MATH 23: SYSTEMS OF LINEAR EQUATIONS Systems of Linear Equations In the plane R 2 the general form of the equation of a line is ax + by = c and that the general equation of a plane in R 3 will be we call

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 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

6. Cholesky factorization

6. Cholesky factorization 6. Cholesky factorization EE103 (Fall 2011-12) triangular matrices forward and backward substitution the Cholesky factorization solving Ax = b with A positive definite inverse of a positive definite matrix

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

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

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

13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.

13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions. 3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in three-space, we write a vector in terms

More information

2.5 Gaussian Elimination

2.5 Gaussian Elimination page 150 150 CHAPTER 2 Matrices and Systems of Linear Equations 37 10 the linear algebra package of Maple, the three elementary 20 23 1 row operations are 12 1 swaprow(a,i,j): permute rows i and j 3 3

More information

Similarity and Diagonalization. Similar Matrices

Similarity and Diagonalization. Similar Matrices MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

More information

University of Lille I PC first year list of exercises n 7. Review

University of Lille I PC first year list of exercises n 7. Review University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients

More information

Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Pearson Education, Inc.

Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Pearson Education, Inc. 2 Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Theorem 8: Let A be a square matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true

More information

SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING. Self Study Course

SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING. Self Study Course SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING Self Study Course MODULE 17 MATRICES II Module Topics 1. Inverse of matrix using cofactors 2. Sets of linear equations 3. Solution of sets of linear

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

MATH36001 Background Material 2015

MATH36001 Background Material 2015 MATH3600 Background Material 205 Matrix Algebra Matrices and Vectors An ordered array of mn elements a ij (i =,, m; j =,, n) written in the form a a 2 a n A = a 2 a 22 a 2n a m a m2 a mn is said to be

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

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

We seek a factorization of a square matrix A into the product of two matrices which yields an

We seek a factorization of a square matrix A into the product of two matrices which yields an LU Decompositions We seek a factorization of a square matrix A into the product of two matrices which yields an efficient method for solving the system where A is the coefficient matrix, x is our variable

More information

Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible:

Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Cramer s Rule and the Adjugate Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Theorem [Cramer s Rule] If A is an invertible

More information

1 Orthogonal projections and the approximation

1 Orthogonal projections and the approximation Math 1512 Fall 2010 Notes on least squares approximation Given n data points (x 1, y 1 ),..., (x n, y n ), we would like to find the line L, with an equation of the form y = mx + b, which is the best fit

More information

Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

More information

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1. MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

More information

Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010

Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010 Section 4.1: Systems of Equations Systems of equations A system of equations consists of two or more equations involving two or more variables { ax + by = c dx + ey = f A solution of such a system is an

More information

2.1: Determinants by Cofactor Expansion. Math 214 Chapter 2 Notes and Homework. Evaluate a Determinant by Expanding by Cofactors

2.1: Determinants by Cofactor Expansion. Math 214 Chapter 2 Notes and Homework. Evaluate a Determinant by Expanding by Cofactors 2.1: Determinants by Cofactor Expansion Math 214 Chapter 2 Notes and Homework Determinants The minor M ij of the entry a ij is the determinant of the submatrix obtained from deleting the i th row and the

More information

Chapter 5. The Inverse; Numerical Methods

Chapter 5. The Inverse; Numerical Methods Vector Spaces in Physics 8/6/ Chapter. The nverse Numerical Methods n the Chapter we discussed the solution of systems of simultaneous linear algebraic equations which could be written in the form C -

More information

1.2 Solving a System of Linear Equations

1.2 Solving a System of Linear Equations 1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1. Solving a System of Linear Equations 1..1 Simple Systems - Basic De nitions As noticed above, the general form of a linear system of m equations in n variables

More information

4. MATRICES Matrices

4. MATRICES Matrices 4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:

More information

1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form

1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form 1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and

More information

2.5 Elementary Row Operations and the Determinant

2.5 Elementary Row Operations and the Determinant 2.5 Elementary Row Operations and the Determinant Recall: Let A be a 2 2 matrtix : A = a b. The determinant of A, denoted by det(a) c d or A, is the number ad bc. So for example if A = 2 4, det(a) = 2(5)

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 20, 2007 Abstract This paper gives a definition of the determinant and lists many of its well-known properties Volumes of parallelepipeds are

More information

Solving Systems of Linear Equations

Solving Systems of Linear Equations LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how

More information

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

7.4. The Inverse of a Matrix. Introduction. Prerequisites. Learning Style. Learning Outcomes 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 =. Similarly a square matrix A may have an inverse B = A where AB =

More information

Chapter 5. Matrices. 5.1 Inverses, Part 1

Chapter 5. Matrices. 5.1 Inverses, Part 1 Chapter 5 Matrices The classification result at the end of the previous chapter says that any finite-dimensional vector space looks like a space of column vectors. In the next couple of chapters we re

More information

APPLICATIONS OF MATRICES. Adj A is nothing but the transpose of the co-factor matrix [A ij ] of A.

APPLICATIONS OF MATRICES. Adj A is nothing but the transpose of the co-factor matrix [A ij ] of A. APPLICATIONS OF MATRICES ADJOINT: Let A = [a ij ] be a square matrix of order n. Let Aij be the co-factor of a ij. Then the n th order matrix [A ij ] T is called the adjoint of A. It is denoted by adj

More information

Au = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.

Au = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively. Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry

More information

Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Numerical Analysis Lecture Notes Peter J. Olver 4. Gaussian Elimination In this part, our focus will be on the most basic method for solving linear algebraic systems, known as Gaussian Elimination in honor

More information