Introduction to Matrices for Engineers


 Diana Boyd
 2 years ago
 Views:
Transcription
1 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 The above matrix is a 4 matrix, i.e. it has three columns and four rows. 1.1 Wh use Matrices? We use matrices in mathematics and engineering because often we need to deal with several variables at once eg the coordinates of a point in the plane are written x, or in space as x,, z and these are often written as column matrices in the form: x and x z It turns out that man operations that are needed to be performed on coordinates of points are linear operations and so can be organized in terms of rectangular arras of numbers, matrices. Then we find that matrices themselves can under certain conditions be added, subtracted and multiplied so that there arises a whole new set of algebraic rules for their manipulation. In general, an n m matrix A looks like: A a 11 a 1 a 1... a 1,m 1 a 1,m a 1 a a... a,m 1 a,m a 1 a a... a,m 1 a,m a n 1,1 a n 1, a n 1,... a n 1,m 1 a n 1,m a n,1 a n, a n,... a n,m 1 a n,m Here, the entries are denoted a ij ; e.g. in Example we have a 11 1, a, etc. Capital letters are usuall used for the matrix itself. 1. Dimension In the above matrix A, the numbers n and m are called the dimensions of A. Based on original lecture notes on matrices b Lewis Pirnie 1
2 Introduction to Matrices 1. Addition It is possible to add two matrices together, but onl if the have the same dimensions. In which case we simpl add the corresponding entries: If two matrices don t have the same size dimensions then the can t be added, or we sa the sum is not defined. 1.4 Example is undefined Multipling Matrices.1 Wh should we want to? We can motivate the process b looking at rotations of points in the plane. Consider the point P on the xaxis at x x, 0, so it has coordinates x 0 Now, let P be the point obtained b rotating P round the origin in an anticlockwise direction through angle θ, keeping the distance from the origin which is actuall the square root of the sum of squares of the coordinates constant. What are the coordinates of the point P? It is not difficult trigonometr to work out that these are: x cos θ x sin θ Next consider the point Q on the axis with coordinates 0 and rotate this point round the origin in an anticlockwise direction through angle θ, keeping the distance from the origin constant. We find that the new coordinates are: sin θ cos θ Now the paoff, it is actuall a matrix that does the rotating here because the operation is linear on the coordinate components. We can express the rotation of an point with coordinates x, as the following matrix equation: cos θ sin θ sin θ cos θ x x cos θ sin θ x sin θ + cos θ Can ou guess what will be the matrix for rotation in a clockwise direction?
3 C.T.J. Dodson. Exercise on Trigonometr 1. Three common right angled triangles can be used to obtain the following values for the trigonometric functions: 0 π 6 45 π 4 60 π sin π 6 1 sin π 4 1 sin π cos π 6 cos π 4 1 cos π 1 tan π 6 1 tan π 4 1 tan π. Obtain the rotation matrices explicitl for rotations of θ ±0, ±45, ±60, ±90, ±180.. Rules When multipling matrices, keep the following in mind: la the first row of the first matrix on top of the first column of the second matrix; onl if the are both of the same size can ou proceed. The rule for multipling is: go across the first matrix, and down the second matrix, multipling the corresponding entries, and adding the products. This new number goes in the new matrix in position of the row of the first matrix, and the column of the second matrix. For example: Smbolicall, if we have the matrices A and B as follows: A a 11 a 1 a 1... a 1,m a 1 a a... a,m a 1 a a... a,m a n 1,1 a n 1, a n 1,... a n 1,m a n,1 a n, a n,... a n,m, B b 11 b 1 b 1... b 1,q b 1 b b... b,q b 1 b b... b,q b p 1,1 b p 1, b p 1, b p 1,q 1 b p 1,q b p,1 b p, b p,... b p,q then the product AB is given b: m i1 a m 1ib i1 i1 a m 1ib i... i1 a m 1ib iq i1 a m ib i1 i1 a m ib i... i1 a ib iq m i1 a m nib i i1 a m nib i... i1 a nib iq where m i1 a 1ib i1 stands for a 11 b 11 + a 1 b 1 + a 1 b a 1n b n1, etc. Note that we must have m p, i.e. the number os columns in the first matrix must equal the number of rows of the second; otherwise, we sa the product is undefined.
4 4 Introduction to Matrices.4 Example We multipl the following matrices: 1 0 i ii is undefined iii iv v vi We see from the examples: i Product of matrix with a matrix is a matrix. iii Product of matrix with a matrix is a matrix. iv Product of matrix with a 4 matrix is a 4 matrix. v Product of matrix with a 1 matrix is a 1 matrix. vi Product of matrix with a 4 matrix is a 4 matrix. Note that the two middle numbers must be the same if the product is defined; and then the dimensions of the answer is just the two outer numbers. Thus, the product of an n m matrix with a m q matrix is an n q matrix. Scalar Multiplication There is another tpe of multiplication involving matrices called scalar multiplication. This means just multipling each entr of the matrix b a number. For example: Rules There are some rules which matrix addition and multiplication obe: Associative A + B + C A + B + C Commutative A + B B + A
5 C.T.J. Dodson 5 Distributive Distributive Associative Non Commutative Moving Constants AB + C AB + AC A + BC AC + BC ABC ABC AB BA usuall AλB λab Assuming that the sums and products are defined in all cases.. Example Let A , B 1 , C Consider the following: AB BA Now, AB BA, and we see that two matrices are not the same if the are multiplied the other wa around. Also consider: AC BC AB + BC A + B A + BC and we notice that A + BC AC + BC as required Transpose Another operation on matrices is the transpose. This just reverses the rows and columns, or equivalentl, reflects the matrix along the leading diagonal. The transpose of A is normall written A t thus a 11 a 1 a 1... a 1,m 1 a 1,m a 1 a a... a,m 1 a,m A a 1 a a... a,m 1 a,m , a n 1,1 a n 1, a n 1,... a n 1,m 1 a n 1,m a n,1 a n, a n,... a n,m 1 a n,m
6 6 Introduction to Matrices A t a 11 a 1 a 1... a m 1,1 a m,1 a 1 a a... a m 1, a m, a 1 a a... a m 1, a m, a 1,n 1 a,n 1 a,n 1... a m 1,n 1 a m,n 1 a 1,n a,n a,n... a m 1,n a m,n Note that the transpose of a n m matrix is a m n matrix. 4.1 Example As an example of the transpose: A , A t Square Matrices Given a number n, n n matrices have ver special properties. Note that if we have two n n matrices, the product is defined and will also be an n n matrix. 5.1 The Identit Matrix There also exists a special matrix, known as the identit, I n n : I , I , I , etc. which has 1 s on the main diagonal, and 0 s everwhere else. This matrix has the propert that, given an n n matrix A: AI n n A and I n n A A i.e. multipling b the identit on either side doesn t change the matrix. This is similar to the propert of 1 when multipling numbers. We usuall abbreviate I n n to just I when it s obvious what n is. 5. Example Let A 1 1 0, I I 0 1 Now, we check the properties of the identit: as required. AI IA
7 C.T.J. Dodson 6 Determinants One of the most important properties of square matrices is the determinant. This is a number obtained from the entries. 6.1 Determinant of a Matrix a b Let A. Then, the determinant of A, denoted det A or A is given b ad bc. c d 6. Example det det Before we go on to larger matrices, we need to define minors. 6. Minors Let A be the n n matrix A a 11 a 1 a 1... a 1,m 1 a 1,m a 1 a a... a,m 1 a,m a 1 a a... a,m 1 a,m a n 1,1 a n 1, a n 1,... a n 1,m 1 a n 1,m a n,1 a n, a n,... a n,m 1 a n,m Then, the minor m ij, for each i, j, is the determinant of the n 1 n 1 matrix obtained b deleting the i th row and the j th column. For example, in the above notation: 6.4 Example m 11 det m 1 det We compute all the minors of A a a... a,m 1 a,m a a... a,m 1 a,m a n 1, a n 1,... a n 1,m 1 a n 1,m a n, a n,... a n,m 1 a n,m a 1 a 1... a 1,m 1 a 1,m a a... a,m 1 a,m a n 1, a n 1,... a n 1,m 1 a n 1,m a n, a n,... a n,m 1 a n,m m m 1 m m m m
8 8 Introduction to Matrices m m m Minors and Cofactors The numbers called cofactors are almost the same as minors, except some have a minus sign in accordance with the following pattern: The best wa to remember this is as an alternating or chessboard pattern. The cofactors from the previous example are: c 11 m 11 8 c 1 m 1 15 c 1 m 1 0 c 1 m 1 c m 9 c m 5 c 1 m 1 c m 6 c m 8 Determinant of a Matrix In order to calculate the determinant of a matrix, choose an row or column. Then, multipl each entr b its corresponding cofactor, and add the three products. This gives the determinant..1 Example Letting A 0 4 as before, we compute the determinant using the top row: det A a 11 c 11 + a 1 c 1 + a 1 c Suppose, we use the second column instead: det A a 1 c 1 + a c + a c It doesn t matter which row or column is used, but the top row is normal. Note that it is not necessar to work out all the minors or cofactors, just three.. Example Let B We compute the determinant of B: det
9 C.T.J. Dodson 9 8 Determinant of an n n Matrix The procedure for larger matrices is exactl the same as for a matrix: choose a row or column, multipl the entr b the corresponding cofactor, and add them up. But of course each minor is itself the determinant of an n 1 n 1 matrix, so for example, in a 4 4 determinant, it is first necessar to do four determinants quite a lot of work! 9 Inverses Let A be an n n matrix, and let I be the n n identit matrix. Sometimes, there exists a matrix A 1 called the inverse of A with the propert: A A 1 I A 1 A In this section, we demonstrate a method for finding inverses. 9.1 Inverse of a Matrix a b Let A. Then, the inverse of A, A c d 1 is given b: A 1 1 d b ad bc c a To check, we multipl: A 1 1 d b a b 1 da bc db bd A ad bc c a c d ad bc ca + ac cb + ad 1 ad bc ad bc 0 ad bc 0 1 In a similar fashion we could show that A A 1 I. Of course, the inverse could also be written A 1 1 d b det A c a Note, that if det A 0, then we have a division b zero, which we can t do. In this situation there is no inverse of A. 9. Inverse of and higher Matrices Recall the definition of a minor from Section 4..: given an n n matrix A, the minor m ij is the determinant of the n 1 n 1 matrix obtained b omitting the i th row and the j th column. 9. Example Let A We calculate the minors: m m 1 m m m m m
10 10 Introduction to Matrices Recall also the pattern of + and signs from which we obtain the cofactors: Now, we put the minors into a matrix and change their signs according to the pattern to get the matrix of cofactors: The next stage is take the transpose: and finall we must divide b the determinant, which is, from Example 4..8: A This shows how to calculate the inverse of a matrix. We check the result: A 1 A as required The same procedure works for n n matrices I II III IV Work out the minors. Put in the signs to form the cofactors. Take the transpose. Divide b the determinant. Furthermore, an n n matrix has an inverse if and onl if the determinant is not zero. So, it s a good idea to calculate the determinant first, just to check whether the rest of the procedure is necessar. 10 Linear Sstems We discuss one ver important application of finding inverses of matrices Simultaneous Equations Often, when solving problems in mathematics, we need to solve simultaneous equations, e.g. something like: x + 5x +
11 C.T.J. Dodson 11 from which we would obtain x and 1. The process we have used up until now is a little mess: we combine the equations to tr and eliminate one of the unknown variables. There is a more sstematic wa using matrices. We can write the equations in a slightl different wa: x + 5x + Now we can check that the first matrix is equal to the product: x + 1 x 5x + 5 and so altogether we have a matrix equation: 1 x 5 The next stage is to use the inverse of the matrix, so let s calculate that now. Let A 1 5, then A Now, we take the matrix equation above, and multipl b A x Then, doing the multiplication: x x x 1 and so x and 1, as required. So, provided we can work out the inverse of the matrix of coefficients, we can solve simultaneous equations. 11 Larger Sstems The same thing works with equations in x, and z. Suppose we have x + + z 1 z x + 8z Then, the matrix form is x z 1 Now, we denote the matrix b A, and calculate the inverse of A. The minors are as follows:
12 1 Introduction to Matrices m m m m m m m 110 m m 1 0 Recall the chessboard pattern: So we have the following matrix of cofactors: We can calculate the determinant b taking an row or column, and multipling the original matrix entr b its corresponding cofactor, and then adding let s choose the top row: det A and so the determinant is, and so we will be able to find the inverse. From the matrix of cofactors, we take the transpose, and then divide b the determinant to get A 1 : A Now, we return to solving the simultaneous equations, where we had: 1 x z Multipling both sides on the left b A 1, we have: x z and we know that A 1 A I, so x z x z and so we get x 64, 1 and z. It s a ver good idea to check calculations like this! x + + z z 1 9 x + 8z as required.
13 C.T.J. Dodson 1 1 Appendix 1.1 Scientific Wordprocessing with L A TEX This pdf document with its hperlinks was created using L A TEX which is the standard free mathematical wordprocessing package; more information can be found via the webpage: 1. Computer Algebra Methods The computer algebra package Mathematica can be used to manipulate and invert matrices. See: for beginning Mathematica. Similarl, Maple and Matlab also can be used for working with matrices.
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 information9 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 informationMATH 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 mbyn matrix is a rectangular array of numbers that has m rows and n columns: a 11
More informationThe 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 information12.3 Inverse Matrices
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 3 7 2 5 A 5 7 2 3 If we think of the identit
More information(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 information4. 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 information1 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 informationUNIT 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 informationMATRIX 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 informationHelpsheet. 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( % . 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 informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationMATRIX 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 informationMatrices Worksheet. Adding the results together, using the matrices, gives
Matrices Worksheet This worksheet is designed to help you increase your confidence in handling MATRICES. This worksheet contains both theory and exercises which cover. Introduction. Order, Addition and
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus ndimensional 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 informationUse 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 informationA 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 informationIntroduction 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 informationThe Solution of Linear Simultaneous Equations
Appendix A The Solution of Linear Simultaneous Equations Circuit analysis frequently involves the solution of linear simultaneous equations. Our purpose here is to review the use of determinants to solve
More informationMATH 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 informationMathematics 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 information2.5 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 information2.1: 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 information9 MATRICES AND TRANSFORMATIONS
9 MATRICES AND TRANSFORMATIONS Chapter 9 Matrices and Transformations Objectives After studying this chapter you should be able to handle matrix (and vector) algebra with confidence, and understand the
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
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 informationMatrix Algebra LECTURE 1. Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 = a 11 x 1 + a 12 x 2 + +a 1n x n,
LECTURE 1 Matrix Algebra Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 a 11 x 1 + a 12 x 2 + +a 1n x n, (1) y 2 a 21 x 1 + a 22 x 2 + +a 2n x n, y m a m1 x 1 +a m2 x
More informationChapter 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 informationLecture 2 Matrix Operations
Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrixvector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or
More informationMatrices, transposes, and inverses
Matrices, transposes, and inverses Math 40, Introduction to Linear Algebra Wednesday, February, 202 Matrixvector multiplication: two views st perspective: A x is linear combination of columns of A 2 4
More informationMatrix 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 informationMATH 2030: MATRICES. Av = 3
MATH 030: MATRICES Introduction to Linear Transformations We have seen that we ma describe matrices as smbol with simple algebraic properties like matri multiplication, addition and scalar addition In
More informationLecture 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 informationBasics 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 informationSection V.2: Magnitudes, Directions, and Components of Vectors
Section V.: Magnitudes, Directions, and Components of Vectors Vectors in the plane If we graph a vector in the coordinate plane instead of just a grid, there are a few things to note. Firstl, directions
More informationWe 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 informationInverses 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 informationMathematical Procedures
CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,
More informationLinear 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 informationSection 5: The Jacobian matrix and applications. S1: Motivation S2: Jacobian matrix + differentiability S3: The chain rule S4: Inverse functions
Section 5: The Jacobian matri and applications. S1: Motivation S2: Jacobian matri + differentiabilit S3: The chain rule S4: Inverse functions Images from Thomas calculus b Thomas, Wier, Hass & Giordano,
More informationCofactor 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 informationMathematics Notes for Class 12 chapter 10. Vector Algebra
1 P a g e Mathematics Notes for Class 12 chapter 10. Vector Algebra A vector has direction and magnitude both but scalar has only magnitude. Magnitude of a vector a is denoted by a or a. It is nonnegative
More informationMatrices: 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 information13 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 threespace, we write a vector in terms
More informationCalculus 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.rwthaachen.de Lehrstuhl A für Mathematik, RWTH Aachen October 30, 2008 Overview
More informationThe basic unit in matrix algebra is a matrix, generally expressed as: a 11 a 12. a 13 A = a 21 a 22 a 23
(copyright by Scott M Lynch, February 2003) Brief Matrix Algebra Review (Soc 504) Matrix algebra is a form of mathematics that allows compact notation for, and mathematical manipulation of, highdimensional
More informationDefinition 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 informationDeterminants. 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 informationB 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 informationa 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 informationMATH36001 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 informationMatrix 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 informationMath 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 informationMatrices Summary. To add or subtract matrices they must be the same dimensions. Just add or subtract the corresponding numbers.
Matrices Summary To transpose a matrix write the rows as columns. Academic Skills Advice For example: 2 1 A = [ 1 2 1 0 0 9] A T = 4 2 2 1 2 1 1 0 4 0 9 2 To add or subtract matrices they must be the same
More informationTrigonometric Functions and Triangles
Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between
More information7.4. 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 informationMath 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 informationVector Notation: AB represents the vector from point A to point B on a graph. The vector can be computed by B A.
1 Linear Transformations Prepared by: Robin Michelle King A transformation of an object is a change in position or dimension (or both) of the object. The resulting object after the transformation is called
More informationMath 018 Review Sheet v.3
Math 018 Review Sheet v.3 Tyrone Crisp Spring 007 1.1  Slopes and Equations of Lines Slopes: Find slopes of lines using the slope formula m y y 1 x x 1. Positive slope the line slopes up to the right.
More informationEC9A0: Presessional Advanced Mathematics Course
University of Warwick, EC9A0: Presessional Advanced Mathematics Course Peter J. Hammond & Pablo F. Beker 1 of 55 EC9A0: Presessional Advanced Mathematics Course Slides 1: Matrix Algebra Peter J. Hammond
More informationGCE Mathematics (6360) Further Pure unit 4 (MFP4) Textbook
Version 36 klm GCE Mathematics (636) Further Pure unit 4 (MFP4) Textbook The Assessment and Qualifications Alliance (AQA) is a company limited by guarantee registered in England and Wales 364473 and a
More informationrow row row 4
13 Matrices The following notes came from Foundation mathematics (MATH 123) Although matrices are not part of what would normally be considered foundation mathematics, they are one of the first topics
More informationTrigonometry Review Workshop 1
Trigonometr Review Workshop Definitions: Let P(,) be an point (not the origin) on the terminal side of an angle with measure θ and let r be the distance from the origin to P. Then the si trig functions
More informationTopic 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 information1 Symmetries of regular polyhedra
1230, notes 5 1 Symmetries of regular polyhedra Symmetry groups Recall: Group axioms: Suppose that (G, ) is a group and a, b, c are elements of G. Then (i) a b G (ii) (a b) c = a (b c) (iii) There is an
More informationSolving a System of Equations
11 Solving a System of Equations 111 Introduction The previous chapter has shown how to solve an algebraic equation with one variable. However, sometimes there is more than one unknown that must be determined
More informationA 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 informationUNIVERSITY OF WISCONSIN SYSTEM
Name UNIVERSITY OF WISCONSIN SYSTEM MATHEMATICS PRACTICE EXAM Check us out at our website: http://www.testing.wisc.edu/center.html GENERAL INSTRUCTIONS: You will have 90 minutes to complete the mathematics
More informationLectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain
Lectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n realvalued matrix A is said to be an orthogonal
More informationEIGENVALUES 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 information5.3 The Cross Product in R 3
53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or
More informationLecture 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 informationMATHEMATICS CLASS  XII BLUE PRINT  II. (1 Mark) (4 Marks) (6 Marks)
BLUE PRINT  II MATHEMATICS CLASS  XII S.No. Topic VSA SA LA TOTAL ( Mark) (4 Marks) (6 Marks). (a) Relations and Functions 4 () 6 () 0 () (b) Inverse trigonometric Functions. (a) Matrices Determinants
More information15.062 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 informationLecture L3  Vectors, Matrices and Coordinate Transformations
S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3  Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between
More informationHomework: 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 informationComplex Numbers. w = f(z) z. Examples
omple Numbers Geometrical Transformations in the omple Plane For functions of a real variable such as f( sin, g( 2 +2 etc ou are used to illustrating these geometricall, usuall on a cartesian graph. If
More informationSECTION 74 Algebraic Vectors
74 lgebraic Vectors 531 SECTIN 74 lgebraic Vectors From Geometric Vectors to lgebraic Vectors Vector ddition and Scalar Multiplication Unit Vectors lgebraic Properties Static Equilibrium Geometric vectors
More informationSection 5.3. Section 5.3. u m ] l jj. = l jj u j + + l mj u m. v j = [ u 1 u j. l mj
Section 5. l j v j = [ u u j u m ] l jj = l jj u j + + l mj u m. l mj Section 5. 5.. Not orthogonal, the column vectors fail to be perpendicular to each other. 5..2 his matrix is orthogonal. Check that
More informationDeterminants LECTURE Calculating the Area of a Parallelogram. Definition Let A be a 2 2 matrix. A = The determinant of A is the number
LECTURE 13 Determinants 1. Calculating the Area of a Parallelogram Definition 13.1. Let A be a matrix. [ a c b d ] The determinant of A is the number det A) = ad bc Now consider the parallelogram formed
More informationChapter 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 informationCore Maths C3. Revision Notes
Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...
More informationTypical 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 informationSergei Silvestrov, Christopher Engström, Karl Lundengård, Johan Richter, Jonas Österberg. November 13, 2014
Sergei Silvestrov,, Karl Lundengård, Johan Richter, Jonas Österberg November 13, 2014 Analysis Todays lecture: Course overview. Repetition of matrices elementary operations. Repetition of solvability of
More informationChapter 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 information4. 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 informationMatrices 2. Solving Square Systems of Linear Equations; Inverse Matrices
Matrices 2. Solving Square Systems of Linear Equations; Inverse Matrices Solving square systems of linear equations; inverse matrices. Linear algebra is essentially about solving systems of linear equations,
More information1 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 informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationSYSTEMS OF EQUATIONS AND MATRICES WITH THE TI89. by Joseph Collison
SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI89 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 informationMatrices, Determinants and Linear Systems
September 21, 2014 Matrices A matrix A m n is an array of numbers in rows and columns a 11 a 12 a 1n r 1 a 21 a 22 a 2n r 2....... a m1 a m2 a mn r m c 1 c 2 c n We say that the dimension of A is m n (we
More informationA linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form
Section 1.3 Matrix Products A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form (scalar #1)(quantity #1) + (scalar #2)(quantity #2) +...
More informationIdentifying 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 informationwww.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates
Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c
More informationMATHEMATICS FOR ENGINEERS BASIC MATRIX THEORY TUTORIAL 2
MATHEMATICS FO ENGINEES BASIC MATIX THEOY TUTOIAL This is the second of two tutorials on matrix theory. On completion you should be able to do the following. Explain the general method for solving simultaneous
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationSECTION 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 informationAPPLICATIONS OF MATRICES. Adj A is nothing but the transpose of the cofactor matrix [A ij ] of A.
APPLICATIONS OF MATRICES ADJOINT: Let A = [a ij ] be a square matrix of order n. Let Aij be the cofactor of a ij. Then the n th order matrix [A ij ] T is called the adjoint of A. It is denoted by adj
More informationMathQuest: 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 informationMATH REVIEW KIT. Reproduced with permission of the Certified General Accountant Association of Canada.
MATH REVIEW KIT Reproduced with permission of the Certified General Accountant Association of Canada. Copyright 00 by the Certified General Accountant Association of Canada and the UBC Real Estate Division.
More information