Introduction to Matrices and Linear Systems Ch. 3
|
|
- Blaise Griffin
- 5 years ago
- Views:
Transcription
1 Introduction to Matrices and Linear Systems Ch. 3 Doreen De Leon Department of Mathematics, California State University, Fresno June, 5 Basic Matrix Concepts and Operations Section 3.4. Basic Matrix Concepts Definition: A matrix is an array of numbers. Notation: a a a n a a a n A = (a ij ) = a m a m a mn A is an m n matrix, where m is the number of rows and n is the number of columns. If m = n, A is called a square matrix. If A is a square matrix, the diagonal of A is the set of entries of the form a ii. The diagonal is only defined from the top left to the bottom right of the matrix. Applications Incidence Matrices: Matrices can be used to characterize connections in electrical networks, in nets of roads, in production processes, etc. as follows.. Nodal Incidence Matrix. The newtwork below consists of six branches (connections) and four nodes (points where two or more branches come toghether). One node is the reference node (gournded node, whose voltage is zero). We number the other nodes and number and direct the branches. This is done arbitrarily. The network can now be described by a matrix A = (a jk ), where + if branch k leaves node j a jk = if branch k enters node j if branch k does not touch node j.
2 Example: A is called the nodal incidence matrix of the network. For the network in the example, A =.. Mesh Incidence Matrix. A network can also be characterized by the mesh incidence matrix M = (m jk ), where + if branch k is in mesh j and has the same orientation m jk = if branch k is in mesh j and has the opposite orientation if branch k is not in mesh j, and a mesh is a loop with no branch in its interior (or in its exterior). Here, the meshes are numbered and directed (oriented) in an arbitrary fashion. We can label the meshes for the network above as follows:
3 Then, the mesh incidence matrix is M =. Example Coefficient Matrices: Given the system of equations 3x y + 5z = x + 4y = the coefficient matrix is ( Vectors: Vectors come in two flavors, ). (column) vector a matrix having only one column, and row vector a matrix with only one row. A (column) vector has the form Example: A row vector is of the form Example: v v v =.. v = v n v = ( v v v n ). u = ( ). Note: Often, a column vector is associated with a point in space and is written as, e.g., v = (5, 4, 3, ). 3
4 Special matrices: upper triangular a square matrix having all zeros below the diagonal. Any of the other entries may be zero. The basic form is a a a n a a n a nn Example: 3. lower triangular a square matrix having all zeros above the diagonal. Any of the other entries may be zero. The basic form is a a a a n a n a nn Example: ( ). 3 diagonal a square matrix having all zeros above and below the diagonal. Any of the entries on the diagonal may be zero. The basic form is a a a nn Examples: the identity matrix I (the diagonal matrix having all ones on the diagonal); the zero matrix (the matrix having only zeros). Example: the 3 3 identity matrix. 4
5 Example: the 3 3 zero matrix.. Matrix Operations.. Transposition Definition: The transpose of a matrix A, denoted A T, is defined by swapping the rows and columns. In mathematical notation, it is defined as a ij a ji. Examples: Definitions: v = v T = ( 3 ) 3 ( ) 3 A = A 4 T = 3 4 symmetric A matrix A is symmetric if A T = A. Example: A = skew-symmetric A matrix A is skew-symmetric if A T = A. Example: A = Equality of Matrices A = (a ij ) and B = (b ij ) are equal if and only if i) A and B have the same size, and ii) a ij = b ij for all i, j. 5
6 ..3 Matrix Addition/Subtraction Matrix addition and subtraction are defined only for matrices of the same size. The sum (difference) is found by adding (subtracting) the corresponding entries. Example: A = 3 4, B = A + B = = ( ) Scalar Multiplication Let c be a scalar and A an m n matrix. Then ca is found by multiplying each entry in A by c. (So, A is found by taking the negative of each entry.) Example: A = ( ), c = = ca = A = 3 4 ( ) = 3 4 ( ) Some laws: Here, denotes the zero matrix (i.e., the matrix whose entries are all zero). A and B are matrices and c, d, and k are scalars. a) A + B = B + A b) A + = A c) A + ( A) = d) c(a + B) = ca + cb, (c + d)a = ca + da e) (ck)a = c(ka) Transposition laws i) (A + B) T = A T + B T ii) (ca) T = ca T 6
7 ..5 Matrix Multiplication Definition: The product C = AB of an m n matrix A and an r p matrix B is defined if and only if r = n, i.e., the number of rows of B must equal the number of columns of A. C is then the m p matrix with entries n c ij = a il b lj = a i b j + a i b j + + a in b nj. l= Idea: Take the dot product of each row i in A with each column j in B and place the results in the corresponding position of C (the ij th position). Some Applications of Matrix Multiplication. Exercise for Weight Loss (requires matrix-vector product). Suppose that in a diet by exercise program, a person weighing 85 pounds burns 35 calories per hour in walking (3 mph), 5 in bicycling (3 mph), and 95 in jogging (6.6 mph). Bill, weighing 85 lb, plans to exercise the number of hours each day given in matrix A. Vector v contains the calories burned in each exercise. Thus, the total calories burned each day can be determined by finding the matrix-vector product Av. and A = W B J v = MON WED FRI SAT. Computer Production (requires matrix-matrix product). Suppose a computer production company produces two major models of computers, PC86 and PC86. The matrix A shows the cost per computer (in thousands of dollars) and matrix B shows the production figures for each quarter for the year (in multiples of, units). The product of the two matrices will show shareholders the cost per quarter (in millions of dollars) for raw material, labor, and miscellaneous. and A = PC86 PC B = Quarter 3 4 ( ) Raw Components Labor Miscellaneous PC86 PC86.
8 Examples: ( ) 5 ) A =, v = ( ) Av = ( ) = 9 Note: va is not defined. 3 ) A = 4 5 6, B = 3 3 AB = () + ( ) + 3() () + ( ) + 3() = 4() + 5( ) + 6() 4() + 5( ) + 6() () + ( )( ) + ( 3)() () + ( )( ) + ( 3)() 4 = 5 4 Note: BA is not defined. WARNING: Note that ) AB BA in general. ) AB = does not necessarily imply (a) A = (b) B = (c) BA = 3) AC = AD does not necessarily imply C = D. Other properties of matrix products: In the following, A, B, and C are matrices and k is a scalar. i) (ka)b = k(ab) 8
9 ii) A(BC) = (AB)C iii) (A + B)C = AC + BC iv) C(A + B) = CA + CB v) (AB) T = B T A T.3 Linear Systems of Equations and Gaussian Elimination Section 3. Linear systems of equations appear in many applications, e.g., analysis of circuits using Kirchhoff s laws. The goal in this section is to solve linear systems of equations using Gaussian elimination, a method closely related to the standard method of elimination that you probably learned in high school. The difference is that Gaussian elimination involves using matrices and systematically modifying them to obtain a system directly solvable by back substitution (solving the last equation for the last variable and working our way up to the first equation, substituting the solved variables into the remaining equations). Idea: Write a linear system of equations in matrix form. Let A = the coefficient matrix, a x + a x + + a n x n = b a x + a x + + a n x n = b.... a m x + a m x + + a mn x n = b m x x x n b b b m x =., and b =. () Then we can rewrite () in the form Example: Ax = b. Given the system of equations we have 3x + x x 3 = x + x 3 = x + x 3 = x A =, x = x, and b =. x 3 9
10 Multiply to verify that Ax = b is equivalent to the system given. To solve the system in (), we will first define the augmented matrix a a a n b a a a n b à = (A b) =.... a m a m a mn b m Then, we will use Gaussian elimination to solve: Gaussian elimination uses elementary row operations to reduce à to (row) echelon form; then back-substitution is used to solve the system. First, we will define echelon form and then identify the elementary row operations. The elementary row operations are directly related to the procedures used to solve linear systems of equations by the method of elimination that you probably learned in high school. Definition: A matrix is in (row) echelon form if it satisfies the following properties. Every row consisting entirely of zeros lies beneath every row that contains a nonzero element. In each row with a nonzero element, the first nonzero entry (called the leading entry) lies to the right of the leading entry in every preceding row. There are three elementary row operations. They are Multiply one row by a nonzero constant. Exchange two rows. Add a nonzero constant multiple of one row to another. Definition: Matrix A is row-equivalent to matrix B if B can be obtained from A by a sequence of elementary row operations. We care about this because two systems of equations Ax = b and Bx = d have the same solution if their augmented matrices are row-equivalent. Examples: ) Solve x + x x 3 = x + x + x 3 = 3x + 5x x 3 =
11 r r r 3 3 r r 3 r 3 r 3 3r r 3 r 3 r Using back-substitution, we see that x 3 = x + x 3 = = x = x 3 = 3 x + x x 3 = = x = x + x 3 = 4 In vector form, we write 4 x = 3. ) Solve ( ) x + 4x + 5x 3 = 6 x + 3x + 4x 3 = 7 ( ) r r 3 r = x + x 3 = 9 3x + 4x + 5x 3 = 6 Since we have no leading entry for x 3, x 3 is arbitrary. So, ( r 3r ) In vector form, 3) Solve Let x 3 = r = x = 9 r = x = 6 4x 5x 3 6 4(9 r) 5r = r = = + r 3 + r x = 9 r = r 9 + r x x + 3x 3 = 8 x + x + x 3 = 5 5x + 5x + 6x 3 = 3.
12 r r 3 8 r r r 5 r 3 r 3 5r r 3 r 3 r 5 This leaves us with 5x + x 3 = x + x + x 3 = 5 Again, we have no leading entry for x 3, so x 3 is arbitrary. In vector form, Let x 3 = r = x = x 3 + = r = x = 5 x x 3 ( ) r + = 5 r 5 7r = 5 7r 5 x = r+ = 5 r 5 + r ) Solve x + x 3x 3 = 4 3x + x + x 3 = 8 x x + 4x 3 = r 3r r 3 r 5 4 r 3 r 3 r The last row implies = 4, which is false. Therefore, there is no solution..4 Reduced Row Echelon Matrices and Gauss-Jordan Elimination Section Gauss-Jordan Elimination Definition: A matrix is in reduced row echelon form if it has the properties of a matrix in row echelon form and Each leading entry is.
13 Each leading entry is the only nonzero element in the column. Examples: The following matrices are in reduced row echelon form. 3,, and 5 The goal of Gauss-Jordan elimination is to transform a matrix to reduced row echelon form. Theorem. Every matrix is row-equivalent to one and only one reduced row echelon matrix. The advantage of reduced row echelon form: You can easily see the number and type of solutions of a system of equations. Theorem. Every linear system either has a unique solution, no solution, or infinitely many solutions..4. Homogeneous Systems Definition: A linear system of equations is homogeneous if the right hand side is all zeros. In other words, we have a system of the form a x + a x + + a n x n = a x + a x + + a n x n =.... a m x + a m x + + a mn x n = Note: This system of equations always has the trivial solution (x =, x =,... x n = ). Therefore, homogeneous systems either have a unique solution or infinitely many solutions. Examples: ) Solve using Gauss-Jordan elimination x + x x 3 = x + x + x 3 = 3x + 5x x 3 = r r r 3 r r 3 r 3 r 3 3r r r r 3 r 3 +r r r +r 3 r r r r r r 3 3 3
14 Solving, x 3 = x = x = ) Solve using Gauss-Jordan elimination 3 5 r r x + x x 3 = x + x = 3x + 5x x 3 = r r 3 r 3 r 3 3r r r r r 3 r 3 +r r r r This matrix is in reduced row-echelon form. Since there is no leading entry for x 3, x 3 is arbitrary and Let x 3 = r = x = x 3 = r = x = x 3 = r So, x x = x = x 3 r r r = r Theorem 3. The homogoneous system with n n coefficient matrix A has only the trivial solution if and only if A is row-equivalent to the identity matrix..5 Inverse of a Matrix Section 3.5 Motivating Example - Cryptography. Cryptography is the process of encoding and decoding messages. One type of code that is difficult to break makes use of a large invertible matrix to encode a message, called the encoding matrix. The receiver of the message decodes it using the inverse of the encoding matrix. The inverse of the encoding matrix is called the decoding matrix. We can illustrate this method using a 3 3 matrix for simplicity. Suppose that the message is STUDY MATH MORE, 4
15 and the encoding matrix is We assign a number to each letter of the alphabet. For convenience, we will associate each letter with its position in the alphabet; so, A is, B is, etc. Let a space between words be denoted by the number 7. The digital form of our message is then S T U D Y M A T H M O R E Since we are using a 3 3 matrix to encode the message, we will break the digital message up into a sequence of 3 column vectors. Then, we put the message into code by multiplying each of the vectors by the encoding matrix. This is most easily done by putting the column vectors into a single matrix and multiplying: = The columns of this matrix give the encoded message, which is transmitted in the following form: 4, 4,, 47, 5, 99, 43,, 35, 97, 4, 65, 4, 3, 34. To decode the message, the receiver writes this string as a sequence of 3 column vectors and repeats the above procedure using the inverse of the encoding matrix, which in this case is Thus, to decode the message, compute = 5 7 8, which you can recognize as the original matrix formed before the message was encoded. 5
16 The Matrix Inverse Recall that the n n identity matrix is the diagonal matrix having ones on the diagonal, I =.... Note that AI = IA = A. Definition: The square matrix A is invertible if there exists a matrix B such that B is an inverse matrix of A. AB = BA = I. Theorem 4. If A is invertible, its inverse is unique. The inverse is denoted by A. Properties of the Matrix Inverse () (A ) = A () (A n ) = (A ) n (3) (AB) = B A How to find A The procedure for finding the inverse of a matrix will be illustrated by example. Example: Find the inverse of A =. 3 5 r r r r 3 r 3 r 3 3r r r 3 3 r r r r r 3 r r r 3 r 3 +r r r +r A =
17 Matrix inverses can be used to solve linear systems of equations. Suppose we wish to solve the system of equaitons Ax = b, where A is invertible. Then, For example, if we have Ax = b = A (Ax) = A b (A A)x = A b Ix = A b x = A b. Ax =, where A is the matrix given in the above example, then 4 x = A = Determinants Section 3.6 Determinants have many applications, some of which we will be studying in this class in Chapters 4 and 5, when we discuss linear independence of functions, and in Chapters 6 and 7, when we use determinants to find eigenvalues and eigenvectors. Recall: ( ) a b A = = det(a) = c d a b c d = ad bc a a a 3 A = a a a 3 = det(a) = a 3 a 3 a 33 a a a 3 a a a 3 a 3 a 3 a 33 a a a a a 3 a 3 = a a a 33 + a a 3 a 3 + a 3 a a 3 a 3 a a 3 a 3 a 3 a a 33 a a Determinants of Larger Matrices Definition: Let A = (a ij ) be an n n matrix. The ijth minor of A, denoted M ij, is the determinant of the (n ) (n ) matrix formed from deleting the ith row and jth column of A. The ijth cofactor of A, denoted A ij is defined by A ij = ( ) i+j M ij. 7
18 Example: Given A = M = M 3 = , = + 4 = = A = ( ) + ( ) = = = 3 = A 3 = ( ) +3 (3) = 3, etc. We can find the determinant of a square matrix using a cofactor expansion. Example: Using a cofactor expansion along the first row, we obtain det(a) = a A + a A + + a n A n. Note: We can use any row or column for this purpose. So, or deta = a i A i + a i A i + + a in A in (cofactor expansion along the ith row) deta = a j A j + a j A j + + a nj A nj (cofactor expansion along the jth column) Idea: Pick the row or column that makes things easiest (e.g., that has the most zeros). Example: Find the determinant of A = For this matrix, it is easiest to use the cofactor expansion along the second or fourth column. We choose the fourth column. 4 deta = ( ) ( ) ( ) = ( ) + ( ) + ( ) = + 6 = 6 Some Properties of Determinants () det(a ) = det(a) 8
19 () det(a T ) = det(a) (3) det(ab) = det(a)det(b) Elementary Row Operations and the Determinant The determinant can also be found by using elementary row operations. First, we need the following theorem, and then we need to discuss how the elementary row operations impact the determinant. Theorem 5. The determinant of an upper (or lower) triangular matrix is the product of its diagonal elements Example: Find det(a) for A = det(a) = = ( )(4)( 3) = 48. If we use elementary row operations to reduce a matrix to row echelon form, we can use this theorem to find its determinant. The following describe how the elementary row operations affect the determinant. () If B is obtained by multiplying a row (or column) of A by a constant k, then det(b) = kdet(a). () If B is obtained by interchanging two rows (or columns) of A, then det(b) = det(a). (3) If B is obtained by adding a constant multiple of one row (or column) of A to another row (or column) of A, then det(b) = det(a). Note: If two rows (or columns) of A are identical, then det(a) =. Note: We can use the determinant to decide if a matrix is invertible. Theorem 6. A matrix A is invertible if and only if its determinant is nonzero. So, if the determinant of a matrix is zero, the matrix is not invertible. Definition: If det(a), then A is nonsingular; otherwise, A is singular. 9
20 Example: Find det(a) for A in the cofactor expansion example by using the method of elimination (i.e., use elementary row operations to transform A to a triangular matrix). r r r 4 r 3 r 3 3r r 3 r 3 r r 4 r 4 4r 3 = r 4 r 4 5 r = r 4 r = 4 r 4 r r 3 8 = = ( ()()( 8) 3 ) 4 = 6
Chapter 2:Determinants. Section 2.1: Determinants by cofactor expansion
Chapter 2:Determinants Section 2.1: Determinants by cofactor expansion [ ] a b Recall: The 2 2 matrix is invertible if ad bc 0. The c d ([ ]) a b function f = ad bc is called the determinant and it associates
More informationLinear Systems and Matrices
Department of Mathematics The Chinese University of Hong Kong 1 System of m linear equations in n unknowns (linear system) a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.......
More informationLinear Algebra. Matrices Operations. Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0.
Matrices Operations Linear Algebra Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0 The rectangular array 1 2 1 4 3 4 2 6 1 3 2 1 in which the
More informationReview for Exam Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions.
Review for Exam. Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions. x + y z = 2 x + 2y + z = 3 x + y + (a 2 5)z = a 2 The augmented matrix for
More informationChapter 7. Linear Algebra: Matrices, Vectors,
Chapter 7. Linear Algebra: Matrices, Vectors, Determinants. Linear Systems Linear algebra includes the theory and application of linear systems of equations, linear transformations, and eigenvalue problems.
More informationChapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations
Chapter 1: Systems of linear equations and matrices Section 1.1: Introduction to systems of linear equations Definition: A linear equation in n variables can be expressed in the form a 1 x 1 + a 2 x 2
More informationGraduate Mathematical Economics Lecture 1
Graduate Mathematical Economics Lecture 1 Yu Ren WISE, Xiamen University September 23, 2012 Outline 1 2 Course Outline ematical techniques used in graduate level economics courses Mathematics for Economists
More informationLinear Algebra: Lecture notes from Kolman and Hill 9th edition.
Linear Algebra: Lecture notes from Kolman and Hill 9th edition Taylan Şengül March 20, 2019 Please let me know of any mistakes in these notes Contents Week 1 1 11 Systems of Linear Equations 1 12 Matrices
More informationLecture 10: Determinants and Cramer s Rule
Lecture 0: Determinants and Cramer s Rule The determinant and its applications. Definition The determinant of a square matrix A, denoted by det(a) or A, is a real number, which is defined as follows. -by-
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS AND MATRICES Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationTopics. Vectors (column matrices): Vector addition and scalar multiplication The matrix of a linear function y Ax The elements of a matrix A : A ij
Topics Vectors (column matrices): Vector addition and scalar multiplication The matrix of a linear function y Ax The elements of a matrix A : A ij or a ij lives in row i and column j Definition of a matrix
More information[ Here 21 is the dot product of (3, 1, 2, 5) with (2, 3, 1, 2), and 31 is the dot product of
. Matrices A matrix is any rectangular array of numbers. For example 3 5 6 4 8 3 3 is 3 4 matrix, i.e. a rectangular array of numbers with three rows four columns. We usually use capital letters for matrices,
More informationLecture Notes in Linear Algebra
Lecture Notes in Linear Algebra Dr. Abdullah Al-Azemi Mathematics Department Kuwait University February 4, 2017 Contents 1 Linear Equations and Matrices 1 1.2 Matrices............................................
More informationUndergraduate Mathematical Economics Lecture 1
Undergraduate Mathematical Economics Lecture 1 Yu Ren WISE, Xiamen University September 15, 2014 Outline 1 Courses Description and Requirement 2 Course Outline ematical techniques used in economics courses
More informationVectors and matrices: matrices (Version 2) This is a very brief summary of my lecture notes.
Vectors and matrices: matrices (Version 2) This is a very brief summary of my lecture notes Matrices and linear equations A matrix is an m-by-n array of numbers A = a 11 a 12 a 13 a 1n a 21 a 22 a 23 a
More informationLemma 8: Suppose the N by N matrix A has the following block upper triangular form:
17 4 Determinants and the Inverse of a Square Matrix In this section, we are going to use our knowledge of determinants and their properties to derive an explicit formula for the inverse of a square matrix
More informationMAC Module 3 Determinants. Learning Objectives. Upon completing this module, you should be able to:
MAC 2 Module Determinants Learning Objectives Upon completing this module, you should be able to:. Determine the minor, cofactor, and adjoint of a matrix. 2. Evaluate the determinant of a matrix by cofactor
More informationFundamentals of Engineering Analysis (650163)
Philadelphia University Faculty of Engineering Communications and Electronics Engineering Fundamentals of Engineering Analysis (6563) Part Dr. Omar R Daoud Matrices: Introduction DEFINITION A matrix is
More informationIntroduction. Vectors and Matrices. Vectors [1] Vectors [2]
Introduction Vectors and Matrices Dr. TGI Fernando 1 2 Data is frequently arranged in arrays, that is, sets whose elements are indexed by one or more subscripts. Vector - one dimensional array Matrix -
More informationDigital Workbook for GRA 6035 Mathematics
Eivind Eriksen Digital Workbook for GRA 6035 Mathematics November 10, 2014 BI Norwegian Business School Contents Part I Lectures in GRA6035 Mathematics 1 Linear Systems and Gaussian Elimination........................
More informationEquality: Two matrices A and B are equal, i.e., A = B if A and B have the same order and the entries of A and B are the same.
Introduction Matrix Operations Matrix: An m n matrix A is an m-by-n array of scalars from a field (for example real numbers) of the form a a a n a a a n A a m a m a mn The order (or size) of A is m n (read
More informationChapter 4 - MATRIX ALGEBRA. ... a 2j... a 2n. a i1 a i2... a ij... a in
Chapter 4 - MATRIX ALGEBRA 4.1. Matrix Operations A a 11 a 12... a 1j... a 1n a 21. a 22.... a 2j... a 2n. a i1 a i2... a ij... a in... a m1 a m2... a mj... a mn The entry in the ith row and the jth column
More informationMath 4A Notes. Written by Victoria Kala Last updated June 11, 2017
Math 4A Notes Written by Victoria Kala vtkala@math.ucsb.edu Last updated June 11, 2017 Systems of Linear Equations A linear equation is an equation that can be written in the form a 1 x 1 + a 2 x 2 +...
More informationChapter 2: Matrices and Linear Systems
Chapter 2: Matrices and Linear Systems Paul Pearson Outline Matrices Linear systems Row operations Inverses Determinants Matrices Definition An m n matrix A = (a ij ) is a rectangular array of real numbers
More informationMATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam. Topics
MATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam This study sheet will not be allowed during the test Books and notes will not be allowed during the test Calculators and cell phones
More informationMatrices and Linear Algebra
Contents Quantitative methods for Economics and Business University of Ferrara Academic year 2017-2018 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2
More informationA matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and
Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.
More informationA matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and
Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.
More informationMATH2210 Notebook 2 Spring 2018
MATH2210 Notebook 2 Spring 2018 prepared by Professor Jenny Baglivo c Copyright 2009 2018 by Jenny A. Baglivo. All Rights Reserved. 2 MATH2210 Notebook 2 3 2.1 Matrices and Their Operations................................
More informationLinear Algebra (part 1) : Matrices and Systems of Linear Equations (by Evan Dummit, 2016, v. 2.02)
Linear Algebra (part ) : Matrices and Systems of Linear Equations (by Evan Dummit, 206, v 202) Contents 2 Matrices and Systems of Linear Equations 2 Systems of Linear Equations 2 Elimination, Matrix Formulation
More informationProperties of the Determinant Function
Properties of the Determinant Function MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Overview Today s discussion will illuminate some of the properties of the determinant:
More informationFinite Mathematics Chapter 2. where a, b, c, d, h, and k are real numbers and neither a and b nor c and d are both zero.
Finite Mathematics Chapter 2 Section 2.1 Systems of Linear Equations: An Introduction Systems of Equations Recall that a system of two linear equations in two variables may be written in the general form
More informationElementary Linear Algebra
Elementary Linear Algebra Linear algebra is the study of; linear sets of equations and their transformation properties. Linear algebra allows the analysis of; rotations in space, least squares fitting,
More informationFormula for the inverse matrix. Cramer s rule. Review: 3 3 determinants can be computed expanding by any row or column
Math 20F Linear Algebra Lecture 18 1 Determinants, n n Review: The 3 3 case Slide 1 Determinants n n (Expansions by rows and columns Relation with Gauss elimination matrices: Properties) Formula for the
More informationElementary Linear Algebra
Matrices J MUSCAT Elementary Linear Algebra Matrices Definition Dr J Muscat 2002 A matrix is a rectangular array of numbers, arranged in rows and columns a a 2 a 3 a n a 2 a 22 a 23 a 2n A = a m a mn We
More informationDeterminants Chapter 3 of Lay
Determinants Chapter of Lay Dr. Doreen De Leon Math 152, Fall 201 1 Introduction to Determinants Section.1 of Lay Given a square matrix A = [a ij, the determinant of A is denoted by det A or a 11 a 1j
More informationChapter 3. Determinants and Eigenvalues
Chapter 3. Determinants and Eigenvalues 3.1. Determinants With each square matrix we can associate a real number called the determinant of the matrix. Determinants have important applications to the theory
More informationIntroduction to Determinants
Introduction to Determinants For any square matrix of order 2, we have found a necessary and sufficient condition for invertibility. Indeed, consider the matrix The matrix A is invertible if and only if.
More informationMath Linear Algebra Final Exam Review Sheet
Math 15-1 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a component-wise operation. Two vectors v and w may be added together as long as they contain the same number n of
More informationChapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form
Chapter 5. Linear Algebra A linear (algebraic) 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 and b are real numbers. 1
More informationAPPENDIX: MATHEMATICAL INDUCTION AND OTHER FORMS OF PROOF
ELEMENTARY LINEAR ALGEBRA WORKBOOK/FOR USE WITH RON LARSON S TEXTBOOK ELEMENTARY LINEAR ALGEBRA CREATED BY SHANNON MARTIN MYERS APPENDIX: MATHEMATICAL INDUCTION AND OTHER FORMS OF PROOF When you are done
More informationLecture Summaries for Linear Algebra M51A
These lecture summaries may also be viewed online by clicking the L icon at the top right of any lecture screen. Lecture Summaries for Linear Algebra M51A refers to the section in the textbook. Lecture
More information1 - Systems of Linear Equations
1 - Systems of Linear Equations 1.1 Introduction to Systems of Linear Equations Almost every problem in linear algebra will involve solving a system of equations. ü LINEAR EQUATIONS IN n VARIABLES We are
More informationLINEAR SYSTEMS, MATRICES, AND VECTORS
ELEMENTARY LINEAR ALGEBRA WORKBOOK CREATED BY SHANNON MARTIN MYERS LINEAR SYSTEMS, MATRICES, AND VECTORS Now that I ve been teaching Linear Algebra for a few years, I thought it would be great to integrate
More informationLinear Algebra M1 - FIB. Contents: 5. Matrices, systems of linear equations and determinants 6. Vector space 7. Linear maps 8.
Linear Algebra M1 - FIB Contents: 5 Matrices, systems of linear equations and determinants 6 Vector space 7 Linear maps 8 Diagonalization Anna de Mier Montserrat Maureso Dept Matemàtica Aplicada II Translation:
More informationMatrix & Linear Algebra
Matrix & Linear Algebra Jamie Monogan University of Georgia For more information: http://monogan.myweb.uga.edu/teaching/mm/ Jamie Monogan (UGA) Matrix & Linear Algebra 1 / 84 Vectors Vectors Vector: A
More information1 Matrices and Systems of Linear Equations. a 1n a 2n
March 31, 2013 16-1 16. Systems of Linear Equations 1 Matrices and Systems of Linear Equations An m n matrix is an array A = (a ij ) of the form a 11 a 21 a m1 a 1n a 2n... a mn where each a ij is a real
More informationEvaluating Determinants by Row Reduction
Evaluating Determinants by Row Reduction MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Objectives Reduce a matrix to row echelon form and evaluate its determinant.
More informationChapter 4. Determinants
4.2 The Determinant of a Square Matrix 1 Chapter 4. Determinants 4.2 The Determinant of a Square Matrix Note. In this section we define the determinant of an n n matrix. We will do so recursively by defining
More informationChapter 2. Square matrices
Chapter 2. Square matrices Lecture notes for MA1111 P. Karageorgis pete@maths.tcd.ie 1/18 Invertible matrices Definition 2.1 Invertible matrices An n n matrix A is said to be invertible, if there is a
More informationMATH 240 Spring, Chapter 1: Linear Equations and Matrices
MATH 240 Spring, 2006 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.1 1.6, 2.1 2.2, 3.2 3.8, 4.3 4.5, 5.1 5.3, 5.5, 6.1 6.5, 7.1 7.2, 7.4 DEFINITIONS Chapter 1: Linear
More informationLinear Algebra. Vector Calculus
PART B Linear Algebra. Vector Calculus CHAPTER 7 Linear Algebra: Matrices, Vectors, Determinants. Linear Systems CHAPTER 8 Linear Algebra: Matrix Eigenvalue Problems CHAPTER 9 Vector Differential Calculus.
More informationMatrices. In this chapter: matrices, determinants. inverse matrix
Matrices In this chapter: matrices, determinants inverse matrix 1 1.1 Matrices A matrix is a retangular array of numbers. Rows: horizontal lines. A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 a 41 a
More information3 Matrix Algebra. 3.1 Operations on matrices
3 Matrix Algebra A matrix is a rectangular array of numbers; it is of size m n if it has m rows and n columns. A 1 n matrix is a row vector; an m 1 matrix is a column vector. For example: 1 5 3 5 3 5 8
More informationLinear Algebra. The analysis of many models in the social sciences reduces to the study of systems of equations.
POLI 7 - Mathematical and Statistical Foundations Prof S Saiegh Fall Lecture Notes - Class 4 October 4, Linear Algebra The analysis of many models in the social sciences reduces to the study of systems
More informationLinear Algebra: Lecture Notes. Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics NUI Galway
Linear Algebra: Lecture Notes Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics NUI Galway November 6, 23 Contents Systems of Linear Equations 2 Introduction 2 2 Elementary Row
More informationSome Notes on Linear Algebra
Some Notes on Linear Algebra prepared for a first course in differential equations Thomas L Scofield Department of Mathematics and Statistics Calvin College 1998 1 The purpose of these notes is to present
More information1300 Linear Algebra and Vector Geometry
1300 Linear Algebra and Vector Geometry R. Craigen Office: MH 523 Email: craigenr@umanitoba.ca May-June 2017 Matrix Inversion Algorithm One payoff from this theorem: It gives us a way to invert matrices.
More informationMATRICES AND MATRIX OPERATIONS
SIZE OF THE MATRIX is defined by number of rows and columns in the matrix. For the matrix that have m rows and n columns we say the size of the matrix is m x n. If matrix have the same number of rows (n)
More informationMatrix Arithmetic. j=1
An m n matrix is an array A = Matrix Arithmetic a 11 a 12 a 1n a 21 a 22 a 2n a m1 a m2 a mn of real numbers a ij An m n matrix has m rows and n columns a ij is the entry in the i-th row and j-th column
More informationThings we can already do with matrices. Unit II - Matrix arithmetic. Defining the matrix product. Things that fail in matrix arithmetic
Unit II - Matrix arithmetic matrix multiplication matrix inverses elementary matrices finding the inverse of a matrix determinants Unit II - Matrix arithmetic 1 Things we can already do with matrices equality
More information1 Matrices and Systems of Linear Equations
Linear Algebra (part ) : Matrices and Systems of Linear Equations (by Evan Dummit, 207, v 260) Contents Matrices and Systems of Linear Equations Systems of Linear Equations Elimination, Matrix Formulation
More information7.6 The Inverse of a Square Matrix
7.6 The Inverse of a Square Matrix Copyright Cengage Learning. All rights reserved. What You Should Learn Verify that two matrices are inverses of each other. Use Gauss-Jordan elimination to find inverses
More informationECON 186 Class Notes: Linear Algebra
ECON 86 Class Notes: Linear Algebra Jijian Fan Jijian Fan ECON 86 / 27 Singularity and Rank As discussed previously, squareness is a necessary condition for a matrix to be nonsingular (have an inverse).
More information4. Determinants.
4. Determinants 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 2 2 determinant 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 3 3 determinant 4.1.
More informationINSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW MATRICES
1 CHAPTER 4 MATRICES 1 INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW MATRICES 1 Matrices Matrices are of fundamental importance in 2-dimensional and 3-dimensional graphics programming
More informationElementary maths for GMT
Elementary maths for GMT Linear Algebra Part 2: Matrices, Elimination and Determinant m n matrices The system of m linear equations in n variables x 1, x 2,, x n a 11 x 1 + a 12 x 2 + + a 1n x n = b 1
More informationChapter 1 Matrices and Systems of Equations
Chapter 1 Matrices and Systems of Equations System of Linear Equations 1. A linear equation in n unknowns is an equation of the form n i=1 a i x i = b where a 1,..., a n, b R and x 1,..., x n are variables.
More informationChapter 2 Notes, Linear Algebra 5e Lay
Contents.1 Operations with Matrices..................................1.1 Addition and Subtraction.............................1. Multiplication by a scalar............................ 3.1.3 Multiplication
More informationMath Camp Notes: Linear Algebra I
Math Camp Notes: Linear Algebra I Basic Matrix Operations and Properties Consider two n m matrices: a a m A = a n a nm Then the basic matrix operations are as follows: a + b a m + b m A + B = a n + b n
More informationSystems of Linear Equations and Matrices
Chapter 1 Systems of Linear Equations and Matrices System of linear algebraic equations and their solution constitute one of the major topics studied in the course known as linear algebra. In the first
More informationLinear Algebra Primer
Linear Algebra Primer David Doria daviddoria@gmail.com Wednesday 3 rd December, 2008 Contents Why is it called Linear Algebra? 4 2 What is a Matrix? 4 2. Input and Output.....................................
More informationII. Determinant Functions
Supplemental Materials for EE203001 Students II Determinant Functions Chung-Chin Lu Department of Electrical Engineering National Tsing Hua University May 22, 2003 1 Three Axioms for a Determinant Function
More informationMATH 2050 Assignment 8 Fall [10] 1. Find the determinant by reducing to triangular form for the following matrices.
MATH 2050 Assignment 8 Fall 2016 [10] 1. Find the determinant by reducing to triangular form for the following matrices. 0 1 2 (a) A = 2 1 4. ANS: We perform the Gaussian Elimination on A by the following
More informationMatrix Algebra Determinant, Inverse matrix. Matrices. A. Fabretti. Mathematics 2 A.Y. 2015/2016. A. Fabretti Matrices
Matrices A. Fabretti Mathematics 2 A.Y. 2015/2016 Table of contents Matrix Algebra Determinant Inverse Matrix Introduction A matrix is a rectangular array of numbers. The size of a matrix is indicated
More informationLinear Algebra and Matrix Inversion
Jim Lambers MAT 46/56 Spring Semester 29- Lecture 2 Notes These notes correspond to Section 63 in the text Linear Algebra and Matrix Inversion Vector Spaces and Linear Transformations Matrices are much
More informationTopic 15 Notes Jeremy Orloff
Topic 5 Notes Jeremy Orloff 5 Transpose, Inverse, Determinant 5. Goals. Know the definition and be able to compute the inverse of any square matrix using row operations. 2. Know the properties of inverses.
More informationChapter 9: Systems of Equations and Inequalities
Chapter 9: Systems of Equations and Inequalities 9. Systems of Equations Solve the system of equations below. By this we mean, find pair(s) of numbers (x, y) (if possible) that satisfy both equations.
More informationSection 9.2: Matrices.. a m1 a m2 a mn
Section 9.2: Matrices Definition: A matrix is a rectangular array of numbers: a 11 a 12 a 1n a 21 a 22 a 2n A =...... a m1 a m2 a mn In general, a ij denotes the (i, j) entry of A. That is, the entry in
More informationMath 313 Chapter 1 Review
Math 313 Chapter 1 Review Howard Anton, 9th Edition May 2010 Do NOT write on me! Contents 1 1.1 Introduction to Systems of Linear Equations 2 2 1.2 Gaussian Elimination 3 3 1.3 Matrices and Matrix Operations
More informationA FIRST COURSE IN LINEAR ALGEBRA. An Open Text by Ken Kuttler. Matrix Arithmetic
A FIRST COURSE IN LINEAR ALGEBRA An Open Text by Ken Kuttler Matrix Arithmetic Lecture Notes by Karen Seyffarth Adapted by LYRYX SERVICE COURSE SOLUTION Attribution-NonCommercial-ShareAlike (CC BY-NC-SA)
More informationMath 60. Rumbos Spring Solutions to Assignment #17
Math 60. Rumbos Spring 2009 1 Solutions to Assignment #17 a b 1. Prove that if ad bc 0 then the matrix A = is invertible and c d compute A 1. a b Solution: Let A = and assume that ad bc 0. c d First consider
More informationMTH Linear Algebra. Study Guide. Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education
MTH 3 Linear Algebra Study Guide Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education June 3, ii Contents Table of Contents iii Matrix Algebra. Real Life
More informationis a 3 4 matrix. It has 3 rows and 4 columns. The first row is the horizontal row [ ]
Matrices: Definition: An m n matrix, A m n is a rectangular array of numbers with m rows and n columns: a, a, a,n a, a, a,n A m,n =...... a m, a m, a m,n Each a i,j is the entry at the i th row, j th column.
More informationSystems of Linear Equations and Matrices
Chapter 1 Systems of Linear Equations and Matrices System of linear algebraic equations and their solution constitute one of the major topics studied in the course known as linear algebra. In the first
More information7.5 Operations with Matrices. Copyright Cengage Learning. All rights reserved.
7.5 Operations with Matrices Copyright Cengage Learning. All rights reserved. What You Should Learn Decide whether two matrices are equal. Add and subtract matrices and multiply matrices by scalars. Multiply
More informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More informationSection 9.2: Matrices. Definition: A matrix A consists of a rectangular array of numbers, or elements, arranged in m rows and n columns.
Section 9.2: Matrices Definition: A matrix A consists of a rectangular array of numbers, or elements, arranged in m rows and n columns. That is, a 11 a 12 a 1n a 21 a 22 a 2n A =...... a m1 a m2 a mn A
More informationMath 1314 Week #14 Notes
Math 3 Week # Notes Section 5.: A system of equations consists of two or more equations. A solution to a system of equations is a point that satisfies all the equations in the system. In this chapter,
More informationLinear Algebra Highlights
Linear Algebra Highlights Chapter 1 A linear equation in n variables is of the form a 1 x 1 + a 2 x 2 + + a n x n. We can have m equations in n variables, a system of linear equations, which we want to
More informationMethods for Solving Linear Systems Part 2
Methods for Solving Linear Systems Part 2 We have studied the properties of matrices and found out that there are more ways that we can solve Linear Systems. In Section 7.3, we learned that we can use
More informationLinear Algebra. Solving Linear Systems. Copyright 2005, W.R. Winfrey
Copyright 2005, W.R. Winfrey Topics Preliminaries Echelon Form of a Matrix Elementary Matrices; Finding A -1 Equivalent Matrices LU-Factorization Topics Preliminaries Echelon Form of a Matrix Elementary
More informationDeterminants by Cofactor Expansion (III)
Determinants by Cofactor Expansion (III) Comment: (Reminder) If A is an n n matrix, then the determinant of A can be computed as a cofactor expansion along the jth column det(a) = a1j C1j + a2j C2j +...
More information1 Determinants. 1.1 Determinant
1 Determinants [SB], Chapter 9, p.188-196. [SB], Chapter 26, p.719-739. Bellow w ll study the central question: which additional conditions must satisfy a quadratic matrix A to be invertible, that is to
More informationGaussian Elimination and Back Substitution
Jim Lambers MAT 610 Summer Session 2009-10 Lecture 4 Notes These notes correspond to Sections 31 and 32 in the text Gaussian Elimination and Back Substitution The basic idea behind methods for solving
More informationSystems of Linear Equations. By: Tri Atmojo Kusmayadi and Mardiyana Mathematics Education Sebelas Maret University
Systems of Linear Equations By: Tri Atmojo Kusmayadi and Mardiyana Mathematics Education Sebelas Maret University Standard of Competency: Understanding the properties of systems of linear equations, matrices,
More informationa11 a A = : a 21 a 22
Matrices The study of linear systems is facilitated by introducing matrices. Matrix theory provides a convenient language and notation to express many of the ideas concisely, and complicated formulas are
More informationChapter 5: Matrices. Daniel Chan. Semester UNSW. Daniel Chan (UNSW) Chapter 5: Matrices Semester / 33
Chapter 5: Matrices Daniel Chan UNSW Semester 1 2018 Daniel Chan (UNSW) Chapter 5: Matrices Semester 1 2018 1 / 33 In this chapter Matrices were first introduced in the Chinese Nine Chapters on the Mathematical
More informationPOLI270 - Linear Algebra
POLI7 - Linear Algebra Septemer 8th Basics a x + a x +... + a n x n b () is the linear form where a, b are parameters and x n are variables. For a given equation such as x +x you only need a variable and
More informationPresentation by: H. Sarper. Chapter 2 - Learning Objectives
Chapter Basic Linear lgebra to accompany Introduction to Mathematical Programming Operations Research, Volume, th edition, by Wayne L. Winston and Munirpallam Venkataramanan Presentation by: H. Sarper
More information