Matrix Multiplication

Size: px
Start display at page:

Download "Matrix Multiplication"

Transcription

1 3.2 Matrix Algebra

2 Matrix Multiplication Example Foxboro Stadium has three main concession stands, located behind the south, north and west stands. The top-selling items are peanuts, hot dogs and soda. Sales for the season opener are recorded in the first matrix below, and the prices (in dollars) of the three items are given in the second matrix. South North West Peanuts Hot Dogs Soda

3 Matrix Multiplication Example Foxboro Stadium has three main concession stands, located behind the south, north and west stands. The top-selling items are peanuts, hot dogs and soda. Sales for the season opener are recorded in the first matrix below, and the prices (in dollars) of the three items are given in the second matrix. South North West Peanuts Hot Dogs Soda How can we find the total sales from the south stands?

4 Matrix Multiplication Example Foxboro Stadium has three main concession stands, located behind the south, north and west stands. The top-selling items are peanuts, hot dogs and soda. Sales for the season opener are recorded in the first matrix below, and the prices (in dollars) of the three items are given in the second matrix. South North West Peanuts Hot Dogs Soda How can we find the total sales from the south stands? 120(2.00) + 250(3.00) + 305(2.75) = $

5 Matrix Multiplication Similarly, for the north and west stands, respectively, we get 207(2.00) + 140(3.00) + 419(2.75) = $ and 39(2.00) + 120(3.00) + 190(2.75) = $940.50

6 Matrix Multiplication Similarly, for the north and west stands, respectively, we get and 207(2.00) + 140(3.00) + 419(2.75) = $ (2.00) + 120(3.00) + 190(2.75) = $ We can arrive at this, using matrix multiplication, where the system would look like =

7 Matrix Multiplication Definition If A = [a ij ] is an m n matrix and B = [b ij ] is an n p matrix, then the product AB is an m p matrix AB = [c ij ] where n c ij = a ik b kj = a i1 b 1j + a i2 b 2j a in b nj k=1

8 Matrix Multiplication Definition If A = [a ij ] is an m n matrix and B = [b ij ] is an n p matrix, then the product AB is an m p matrix AB = [c ij ] where n c ij = a ik b kj = a i1 b 1j + a i2 b 2j a in b nj k=1 This is a fancy way of saying that the i, j position in the answer matrix is the dot product of the i th row of the first matrix and the j th column of the second matrix.

9 Matrix Multiplication Definition If A = [a ij ] is an m n matrix and B = [b ij ] is an n p matrix, then the product AB is an m p matrix AB = [c ij ] where n c ij = a ik b kj = a i1 b 1j + a i2 b 2j a in b nj k=1 This is a fancy way of saying that the i, j position in the answer matrix is the dot product of the i th row of the first matrix and the j th column of the second matrix. We also have to make sure that the sizes of the matrices are appropriate for multiplying matrices.

10 Matrix Multiplication Example Example Find the product AB, where 1 3 A = 4 2 and B = 5 0 [ 3 ] 2 4 1

11 Matrix Multiplication Example Example Find the product AB, where 1 3 A = 4 2 and B = 5 0 [ 3 ] [ ] c c 12 = c c c 31 c 32

12 Matrix Multiplication Example 1 3 [ ] c c 12 = c c c 31 c 32

13 Matrix Multiplication Example 1 3 [ ] c c 12 = c c c 31 c 32 c 11 = ( 1)( 3) + 3( 4) = 9

14 Matrix Multiplication Example 1 3 [ ] c c 12 = c c c 31 c 32 c 11 = ( 1)( 3) + 3( 4) = [ ] 9 c = c c c 31 c 32

15 Matrix Multiplication Example 1 3 [ ] 9 c = c c c 31 c 32

16 Matrix Multiplication Example 1 3 [ ] 9 c = c c c 31 c 32 c 12 = 1(2) + 3(1) = 1

17 Matrix Multiplication Example 1 3 [ ] 9 c = c c c 31 c 32 c 12 = 1(2) + 3(1) = [ ] = c c c 31 c 32

18 Matrix Multiplication Example 1 3 [ ] = c c c 31 c 32

19 Matrix Multiplication Example 1 3 [ ] = c c c 31 c 32 c 31 = 5( 3) + 0( 4) = 15

20 Matrix Multiplication Example 1 3 [ ] = c c c 31 c 32 c 31 = 5( 3) + 0( 4) = [ ] = 4 1 c 21 c c 32

21 Matrix Multiplication Example Continuing, we get [ ] =

22 Matrix Multiplication Example [ ] =

23 Matrix Multiplication Example [ ] = [ 5 7 ]

24 Matrix Multiplication Example [ ] = [ 5 7 ] Example [ ] [ ] =

25 Matrix Multiplication Example [ ] = [ 5 7 ] Example [ ] [ ] = [ 3 ] 4 2 5

26 Matrix Multiplication Example [ ] = [ 5 7 ] Example [ ] [ ] = [ 3 ] What do we call the second matrix?

27 Matrix Multiplication Example [ ] [ ] =

28 Matrix Multiplication Example [ ] [ ] = [ ]

29 Matrix Multiplication Example [ ] [ ] = [ ] What is the relationship between these two matrices?

30 Matrix Multiplication Example [ ] [ ] = [ ] What is the relationship between these two matrices? Example [ ] = 1

31 Matrix Multiplication Example [ ] [ ] = [ ] What is the relationship between these two matrices? Example [ ] = [ 1 ] 1

32 Matrix Multiplication Example 2 1 [ ] = 1

33 Matrix Multiplication Example 2 1 [ ] =

34 Commutativity (and the lack of) Note: Matrices are not necessarily commutative. Think about the size of the matrices...

35 Commutativity (and the lack of) Note: Matrices are not necessarily commutative. Think about the size of the matrices... Even if the sizes work, there is no guarantee that there will be equality.

36 Commutativity (and the lack of) Note: Matrices are not necessarily commutative. Think about the size of the matrices... Even if the sizes work, there is no guarantee that there will be equality. Example Find[ the product ] AB and[ BA if ] A = and B =

37 Commutativity (and the lack of) Note: Matrices are not necessarily commutative. Think about the size of the matrices... Even if the sizes work, there is no guarantee that there will be equality. Example Find[ the product ] AB and[ BA if A = and B = [ ] 5 9 AB = 5 17 ]

38 Commutativity (and the lack of) Note: Matrices are not necessarily commutative. Think about the size of the matrices... Even if the sizes work, there is no guarantee that there will be equality. Example Find[ the product ] AB and[ BA if ] A = and B = [ ] [ ] AB = but BA =

39 The Identity Matrix We saw two instances of the identity matrix in the prior examples:

40 The Identity Matrix We saw two instances of the identity matrix in the prior examples: 1 The product of a matrix and the identity is the original matrix

41 The Identity Matrix We saw two instances of the identity matrix in the prior examples: 1 The product of a matrix and the identity is the original matrix 2 The product of a matrix and it s inverse is the identity matrix

42 The Identity Matrix We saw two instances of the identity matrix in the prior examples: 1 The product of a matrix and the identity is the original matrix 2 The product of a matrix and it s inverse is the identity matrix We actually have two identity matrices, depending on the operation.

43 The Identity Matrix We saw two instances of the identity matrix in the prior examples: 1 The product of a matrix and the identity is the original matrix 2 The product of a matrix and it s inverse is the identity matrix We actually have two identity matrices, depending on the operation. Additive Identity For any matrix A M mn, the matrix 0 mn is the additive identity and has the property A + 0 mn = A = 0 mn + A.

44 The Identity Matrix We saw two instances of the identity matrix in the prior examples: 1 The product of a matrix and the identity is the original matrix 2 The product of a matrix and it s inverse is the identity matrix We actually have two identity matrices, depending on the operation. Additive Identity For any matrix A M mn, the matrix 0 mn is the additive identity and has the property A + 0 mn = A = 0 mn + A. We generally refer to this as the zero matrix rather than an identity matrix.

45 The Identity Matrix Multiplicative Identity For any matrix A M n, the matrix I n is the multiplicative identity and has the property AI n = A = I n A.

46 The Identity Matrix Multiplicative Identity For any matrix A M n, the matrix I n is the multiplicative identity and has the property AI n = A = I n A I n =

47 The transpose of a matrix Definition The transpose of a matrix, denoted A T, is the matrix formed from the matrix A = [a ij ] by interchanging the rows and the columns. A T = [a ji ].

48 The transpose of a matrix Definition The transpose of a matrix, denoted A T, is the matrix formed from the matrix A = [a ij ] by interchanging the rows and the columns. A T = [a ji ]. Visually speaking, the transpose of a matrix is a reflection over the main diagonal.

49 The transpose of a matrix Definition The transpose of a matrix, denoted A T, is the matrix formed from the matrix A = [a ij ] by interchanging the rows and the columns. A T = [a ji ]. Visually speaking, the transpose of a matrix is a reflection over the main diagonal. [ ] A = 4 5 6

50 The transpose of a matrix Definition The transpose of a matrix, denoted A T, is the matrix formed from the matrix A = [a ij ] by interchanging the rows and the columns. A T = [a ji ]. Visually speaking, the transpose of a matrix is a reflection over the main diagonal. [ ] A = A T =

51 Properties of Transposes Theorem If A and B are matrices (with sizes such that the given matrix operations are defined) and c is a scalar, then the following properties are true:

52 Properties of Transposes Theorem If A and B are matrices (with sizes such that the given matrix operations are defined) and c is a scalar, then the following properties are true: ( 1 A T ) T = A

53 Properties of Transposes Theorem If A and B are matrices (with sizes such that the given matrix operations are defined) and c is a scalar, then the following properties are true: ( 1 A T ) T = A 2 (A + B) T = A T + B T

54 Properties of Transposes Theorem If A and B are matrices (with sizes such that the given matrix operations are defined) and c is a scalar, then the following properties are true: ( 1 A T ) T = A 2 (A + B) T = A T + B T 3 (ca) T = ca T

55 Properties of Transposes Theorem If A and B are matrices (with sizes such that the given matrix operations are defined) and c is a scalar, then the following properties are true: ( 1 A T ) T = A 2 (A + B) T = A T + B T 3 (ca) T = ca T 4 (AB) T = B T A T

56 Properties of Transposes Theorem If A and B are matrices (with sizes such that the given matrix operations are defined) and c is a scalar, then the following properties are true: ( 1 A T ) T = A 2 (A + B) T = A T + B T 3 (ca) T = ca T 4 (AB) T = B T A T Why do these properties hold?

57 Proof of part 4 (AB) T = B T A T If A is an a b matrix, then B must be a b c matrix.

58 Proof of part 4 (AB) T = B T A T If A is an a b matrix, then B must be a b c matrix. Then, the product AB is an

59 Proof of part 4 (AB) T = B T A T If A is an a b matrix, then B must be a b c matrix. Then, the product AB is an a c matrix, so (AB) T is a c a matrix.

60 Proof of part 4 (AB) T = B T A T If A is an a b matrix, then B must be a b c matrix. Then, the product AB is an a c matrix, so (AB) T is a c a matrix. Furthermore, A T is a b a matrix and B T is a c b matrix, so A T B T cannot exist,

61 Proof of part 4 (AB) T = B T A T If A is an a b matrix, then B must be a b c matrix. Then, the product AB is an a c matrix, so (AB) T is a c a matrix. Furthermore, A T is a b a matrix and B T is a c b matrix, so A T B T cannot exist, but B T A T is a

62 Proof of part 4 (AB) T = B T A T If A is an a b matrix, then B must be a b c matrix. Then, the product AB is an a c matrix, so (AB) T is a c a matrix. Furthermore, A T is a b a matrix and B T is a c b matrix, so A T B T cannot exist, but B T A T is a c a matrix.

63 Proof of part 4 (AB) T = B T A T If A is an a b matrix, then B must be a b c matrix. Then, the product AB is an a c matrix, so (AB) T is a c a matrix. Furthermore, A T is a b a matrix and B T is a c b matrix, so A T B T cannot exist, but B T A T is a c a matrix. Since (AB) T and B T A T have the same size, we now need to show that the entries are the same.

64 Proof of part 4 (AB) T = B T A T If A is an a b matrix, then B must be a b c matrix. Then, the product AB is an a c matrix, so (AB) T is a c a matrix. Furthermore, A T is a b a matrix and B T is a c b matrix, so A T B T cannot exist, but B T A T is a c a matrix. Since (AB) T and B T A T have the same size, we now need to show that the entries are the same. First, note that the i, j th entry of (AB) T is the same as the j, i th entry of AB.

65 Proof of part 4 (AB) T = B T A T If A is an a b matrix, then B must be a b c matrix. Then, the product AB is an a c matrix, so (AB) T is a c a matrix. Furthermore, A T is a b a matrix and B T is a c b matrix, so A T B T cannot exist, but B T A T is a c a matrix. Since (AB) T and B T A T have the same size, we now need to show that the entries are the same. First, note that the i, j th entry of (AB) T is the same as the j, i th entry of AB. Now, the i, j th entry of B T A T is the dot product of the i th row of B T, which is the i th column of B, and the j th column of A T, which is the j th row of A. That is, the i, j th entry of B T A T is the dot product of the j th row of A and i th column of B.

66 Symmetric Matrices Definition A matrix is said to be symmetric if A = A T. Then a ij = a ji for all i j

67 Symmetric Matrices Definition A matrix is said to be symmetric if A = A T. Then a ij = a ji for all i j This implies that all symmetric matrices must be square.

68 Good Will Hunting

69 Good Will Hunting

70 Good Will Hunting 4 A =

71 Good Will Hunting 4 A = A 2 =

Matrices: 2.1 Operations with Matrices

Matrices: 2.1 Operations with Matrices Goals In this chapter and section we study matrix operations: Define matrix addition Define multiplication of matrix by a scalar, to be called scalar multiplication. Define multiplication of two matrices,

More information

Elementary maths for GMT

Elementary 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 information

Mathematics 13: Lecture 10

Mathematics 13: Lecture 10 Mathematics 13: Lecture 10 Matrices Dan Sloughter Furman University January 25, 2008 Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, 2008 1 / 19 Matrices Recall: A matrix is a

More information

ICS 6N Computational Linear Algebra Matrix Algebra

ICS 6N Computational Linear Algebra Matrix Algebra ICS 6N Computational Linear Algebra Matrix Algebra Xiaohui Xie University of California, Irvine xhx@uci.edu February 2, 2017 Xiaohui Xie (UCI) ICS 6N February 2, 2017 1 / 24 Matrix Consider an m n matrix

More information

Phys 201. Matrices and Determinants

Phys 201. Matrices and Determinants Phys 201 Matrices and Determinants 1 1.1 Matrices 1.2 Operations of matrices 1.3 Types of matrices 1.4 Properties of matrices 1.5 Determinants 1.6 Inverse of a 3 3 matrix 2 1.1 Matrices A 2 3 7 =! " 1

More information

Section 9.2: Matrices.. a m1 a m2 a mn

Section 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 information

10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections )

10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections ) c Dr. Igor Zelenko, Fall 2017 1 10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections 7.2-7.4) 1. When each of the functions F 1, F 2,..., F n in right-hand side

More information

CS100: DISCRETE STRUCTURES. Lecture 3 Matrices Ch 3 Pages:

CS100: DISCRETE STRUCTURES. Lecture 3 Matrices Ch 3 Pages: CS100: DISCRETE STRUCTURES Lecture 3 Matrices Ch 3 Pages: 246-262 Matrices 2 Introduction DEFINITION 1: A matrix is a rectangular array of numbers. A matrix with m rows and n columns is called an m x n

More information

2.1 Matrices. 3 5 Solve for the variables in the following matrix equation.

2.1 Matrices. 3 5 Solve for the variables in the following matrix equation. 2.1 Matrices Reminder: A matrix with m rows and n columns has size m x n. (This is also sometimes referred to as the order of the matrix.) The entry in the ith row and jth column of a matrix A is denoted

More information

Kevin James. MTHSC 3110 Section 2.1 Matrix Operations

Kevin James. MTHSC 3110 Section 2.1 Matrix Operations MTHSC 3110 Section 2.1 Matrix Operations Notation Let A be an m n matrix, that is, m rows and n columns. We ll refer to the entries of A by their row and column indices. The entry in the i th row and j

More information

n n matrices The system of m linear equations in n variables x 1, x 2,..., x n can be written as a matrix equation by Ax = b, or in full

n n matrices The system of m linear equations in n variables x 1, x 2,..., x n can be written as a matrix equation by Ax = b, or in full n n matrices Matrices Definitions Diagonal, Identity, and zero matrices Addition Multiplication Transpose and inverse The system of m linear equations in n variables x 1, x 2,..., x n a 11 x 1 + a 12 x

More information

. a m1 a mn. a 1 a 2 a = a n

. a m1 a mn. a 1 a 2 a = a n Biostat 140655, 2008: Matrix Algebra Review 1 Definition: An m n matrix, A m n, is a rectangular array of real numbers with m rows and n columns Element in the i th row and the j th column is denoted by

More information

Matrix Algebra & Elementary Matrices

Matrix Algebra & Elementary Matrices Matrix lgebra & Elementary Matrices To add two matrices, they must have identical dimensions. To multiply them the number of columns of the first must equal the number of rows of the second. The laws below

More information

Section 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. 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 information

Matrix Operations. Linear Combination Vector Algebra Angle Between Vectors Projections and Reflections Equality of matrices, Augmented Matrix

Matrix Operations. Linear Combination Vector Algebra Angle Between Vectors Projections and Reflections Equality of matrices, Augmented Matrix Linear Combination Vector Algebra Angle Between Vectors Projections and Reflections Equality of matrices, Augmented Matrix Matrix Operations Matrix Addition and Matrix Scalar Multiply Matrix Multiply Matrix

More information

MATH 320, WEEK 7: Matrices, Matrix Operations

MATH 320, WEEK 7: Matrices, Matrix Operations MATH 320, WEEK 7: Matrices, Matrix Operations 1 Matrices We have introduced ourselves to the notion of the grid-like coefficient matrix as a short-hand coefficient place-keeper for performing Gaussian

More information

Matrix representation of a linear map

Matrix representation of a linear map Matrix representation of a linear map As before, let e i = (0,..., 0, 1, 0,..., 0) T, with 1 in the i th place and 0 elsewhere, be standard basis vectors. Given linear map f : R n R m we get n column vectors

More information

Matrix Arithmetic. j=1

Matrix 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 information

Materials engineering Collage \\ Ceramic & construction materials department Numerical Analysis \\Third stage by \\ Dalya Hekmat

Materials engineering Collage \\ Ceramic & construction materials department Numerical Analysis \\Third stage by \\ Dalya Hekmat Materials engineering Collage \\ Ceramic & construction materials department Numerical Analysis \\Third stage by \\ Dalya Hekmat Linear Algebra Lecture 2 1.3.7 Matrix Matrix multiplication using Falk s

More information

Matrix representation of a linear map

Matrix representation of a linear map Matrix representation of a linear map As before, let e i = (0,..., 0, 1, 0,..., 0) T, with 1 in the i th place and 0 elsewhere, be standard basis vectors. Given linear map f : R n R m we get n column vectors

More information

Linear Algebra and Matrix Inversion

Linear 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 information

Matrices Gaussian elimination Determinants. Graphics 2009/2010, period 1. Lecture 4: matrices

Matrices Gaussian elimination Determinants. Graphics 2009/2010, period 1. Lecture 4: matrices Graphics 2009/2010, period 1 Lecture 4 Matrices m n matrices Matrices Definitions Diagonal, Identity, and zero matrices Addition Multiplication Transpose and inverse The system of m linear equations in

More information

Matrix Algebra. Matrix Algebra. Chapter 8 - S&B

Matrix Algebra. Matrix Algebra. Chapter 8 - S&B Chapter 8 - S&B Algebraic operations Matrix: The size of a matrix is indicated by the number of its rows and the number of its columns. A matrix with k rows and n columns is called a k n matrix. The number

More information

MATRICES. a m,1 a m,n A =

MATRICES. a m,1 a m,n A = MATRICES Matrices are rectangular arrays of real or complex numbers With them, we define arithmetic operations that are generalizations of those for real and complex numbers The general form a matrix of

More information

Matrix Algebra 2.1 MATRIX OPERATIONS Pearson Education, Inc.

Matrix Algebra 2.1 MATRIX OPERATIONS Pearson Education, Inc. 2 Matrix Algebra 2.1 MATRIX OPERATIONS MATRIX OPERATIONS m n If A is an matrixthat is, a matrix with m rows and n columnsthen the scalar entry in the ith row and jth column of A is denoted by a ij and

More information

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

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 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

Elementary Row Operations on Matrices

Elementary Row Operations on Matrices King Saud University September 17, 018 Table of contents 1 Definition A real matrix is a rectangular array whose entries are real numbers. These numbers are organized on rows and columns. An m n matrix

More information

Matrices. Chapter Definitions and Notations

Matrices. Chapter Definitions and Notations Chapter 3 Matrices 3. Definitions and Notations Matrices are yet another mathematical object. Learning about matrices means learning what they are, how they are represented, the types of operations which

More information

Matrices and Vectors

Matrices and Vectors Matrices and Vectors James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 11, 2013 Outline 1 Matrices and Vectors 2 Vector Details 3 Matrix

More information

Matrices. Math 240 Calculus III. Wednesday, July 10, Summer 2013, Session II. Matrices. Math 240. Definitions and Notation.

Matrices. Math 240 Calculus III. Wednesday, July 10, Summer 2013, Session II. Matrices. Math 240. Definitions and Notation. function Matrices Calculus III Summer 2013, Session II Wednesday, July 10, 2013 Agenda function 1. 2. function function Definition An m n matrix is a rectangular array of numbers arranged in m horizontal

More information

INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW MATRICES

INSTITIÚID 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 information

1 Matrices and matrix algebra

1 Matrices and matrix algebra 1 Matrices and matrix algebra 1.1 Examples of matrices A matrix is a rectangular array of numbers and/or variables. For instance 4 2 0 3 1 A = 5 1.2 0.7 x 3 π 3 4 6 27 is a matrix with 3 rows and 5 columns

More information

A 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 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 information

Section Summary. Sequences. Recurrence Relations. Summations Special Integer Sequences (optional)

Section Summary. Sequences. Recurrence Relations. Summations Special Integer Sequences (optional) Section 2.4 Section Summary Sequences. o Examples: Geometric Progression, Arithmetic Progression Recurrence Relations o Example: Fibonacci Sequence Summations Special Integer Sequences (optional) Sequences

More information

Matrix & Linear Algebra

Matrix & 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 information

CLASS 12 ALGEBRA OF MATRICES

CLASS 12 ALGEBRA OF MATRICES CLASS 12 ALGEBRA OF MATRICES Deepak Sir 9811291604 SHRI SAI MASTERS TUITION CENTER CLASS 12 A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements

More information

Math 360 Linear Algebra Fall Class Notes. a a a a a a. a a a

Math 360 Linear Algebra Fall Class Notes. a a a a a a. a a a Math 360 Linear Algebra Fall 2008 9-10-08 Class Notes Matrices As we have already seen, a matrix is a rectangular array of numbers. If a matrix A has m columns and n rows, we say that its dimensions are

More information

Section 1.6. M N = [a ij b ij ], (1.6.2)

Section 1.6. M N = [a ij b ij ], (1.6.2) The Calculus of Functions of Several Variables Section 16 Operations with Matrices In the previous section we saw the important connection between linear functions and matrices In this section we will

More information

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2018 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing

More information

Instruction: Operations with Matrices ( ) ( ) log 8 log 25 = If the corresponding elements do not equal, then the matrices are not equal.

Instruction: Operations with Matrices ( ) ( ) log 8 log 25 = If the corresponding elements do not equal, then the matrices are not equal. 7 Instruction: Operations with Matrices Two matrices are said to be equal if they have the same size and their corresponding elements are equal. For example, 3 ( ) ( ) ( ) ( ) log 8 log log log3 8 If the

More information

Matrix Arithmetic. a 11 a. A + B = + a m1 a mn. + b. a 11 + b 11 a 1n + b 1n = a m1. b m1 b mn. and scalar multiplication for matrices via.

Matrix Arithmetic. a 11 a. A + B = + a m1 a mn. + b. a 11 + b 11 a 1n + b 1n = a m1. b m1 b mn. and scalar multiplication for matrices via. Matrix Arithmetic There is an arithmetic for matrices that can be viewed as extending the arithmetic we have developed for vectors to the more general setting of rectangular arrays: if A and B are m n

More information

5.1 Introduction to Matrices

5.1 Introduction to Matrices 5.1 Introduction to Matrices Reminder: A matrix with m rows and n columns has size m x n. (This is also sometimes referred to as the order of the matrix.) The entry in the ith row and jth column of a matrix

More information

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing

More information

3. Replace any row by the sum of that row and a constant multiple of any other row.

3. Replace any row by the sum of that row and a constant multiple of any other row. Section. Solution of Linear Systems by Gauss-Jordan Method A matrix is an ordered rectangular array of numbers, letters, symbols or algebraic expressions. A matrix with m rows and n columns has size or

More information

CS 246 Review of Linear Algebra 01/17/19

CS 246 Review of Linear Algebra 01/17/19 1 Linear algebra In this section we will discuss vectors and matrices. We denote the (i, j)th entry of a matrix A as A ij, and the ith entry of a vector as v i. 1.1 Vectors and vector operations A vector

More information

MATH Mathematics for Agriculture II

MATH Mathematics for Agriculture II MATH 10240 Mathematics for Agriculture II Academic year 2018 2019 UCD School of Mathematics and Statistics Contents Chapter 1. Linear Algebra 1 1. Introduction to Matrices 1 2. Matrix Multiplication 3

More information

Lecture 3: Matrix and Matrix Operations

Lecture 3: Matrix and Matrix Operations Lecture 3: Matrix and Matrix Operations Representation, row vector, column vector, element of a matrix. Examples of matrix representations Tables and spreadsheets Scalar-Matrix operation: Scaling a matrix

More information

Prepared by: M. S. KumarSwamy, TGT(Maths) Page

Prepared by: M. S. KumarSwamy, TGT(Maths) Page Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 50 - CHAPTER 3: MATRICES QUICK REVISION (Important Concepts & Formulae) MARKS WEIGHTAGE 03 marks Matrix A matrix is an ordered rectangular array of numbers

More information

Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Numerical Analysis Lecture Notes Peter J Olver 3 Review of Matrix Algebra Vectors and matrices are essential for modern analysis of systems of equations algebrai, differential, functional, etc In this

More information

Math 4377/6308 Advanced Linear Algebra

Math 4377/6308 Advanced Linear Algebra 2.3 Composition Math 4377/6308 Advanced Linear Algebra 2.3 Composition of Linear Transformations Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math4377

More information

Chapter 4 - MATRIX ALGEBRA. ... a 2j... a 2n. a i1 a i2... a ij... a in

Chapter 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 information

MAC Module 2 Systems of Linear Equations and Matrices II. Learning Objectives. Upon completing this module, you should be able to :

MAC Module 2 Systems of Linear Equations and Matrices II. Learning Objectives. Upon completing this module, you should be able to : MAC 0 Module Systems of Linear Equations and Matrices II Learning Objectives Upon completing this module, you should be able to :. Find the inverse of a square matrix.. Determine whether a matrix is invertible..

More information

Introduction. Vectors and Matrices. Vectors [1] Vectors [2]

Introduction. 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 information

MATH 2030: MATRICES ,, a m1 a m2 a mn If the columns of A are the vectors a 1, a 2,...,a n ; A is represented as A 1. .

MATH 2030: MATRICES ,, a m1 a m2 a mn If the columns of A are the vectors a 1, a 2,...,a n ; A is represented as A 1. . MATH 030: MATRICES Matrix Operations We have seen how matrices and the operations on them originated from our study of linear equations In this chapter we study matrices explicitely Definition 01 A matrix

More information

Matrices and Linear Algebra

Matrices 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 information

Matrices BUSINESS MATHEMATICS

Matrices BUSINESS MATHEMATICS Matrices BUSINESS MATHEMATICS 1 CONTENTS Matrices Special matrices Operations with matrices Matrix multipication More operations with matrices Matrix transposition Symmetric matrices Old exam question

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2

MATRIX 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 information

Linear Equations in Linear Algebra

Linear Equations in Linear Algebra 1 Linear Equations in Linear Algebra 1.7 LINEAR INDEPENDENCE LINEAR INDEPENDENCE Definition: An indexed set of vectors {v 1,, v p } in n is said to be linearly independent if the vector equation x x x

More information

DM559 Linear and Integer Programming. Lecture 3 Matrix Operations. Marco Chiarandini

DM559 Linear and Integer Programming. Lecture 3 Matrix Operations. Marco Chiarandini DM559 Linear and Integer Programming Lecture 3 Matrix Operations Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline and 1 2 3 and 4 2 Outline and 1 2

More information

Review of Linear Algebra

Review of Linear Algebra Review of Linear Algebra Definitions An m n (read "m by n") matrix, is a rectangular array of entries, where m is the number of rows and n the number of columns. 2 Definitions (Con t) A is square if m=

More information

Announcements Monday, October 02

Announcements Monday, October 02 Announcements Monday, October 02 Please fill out the mid-semester survey under Quizzes on Canvas WeBWorK 18, 19 are due Wednesday at 11:59pm The quiz on Friday covers 17, 18, and 19 My office is Skiles

More information

Announcements Wednesday, October 10

Announcements Wednesday, October 10 Announcements Wednesday, October 10 The second midterm is on Friday, October 19 That is one week from this Friday The exam covers 35, 36, 37, 39, 41, 42, 43, 44 (through today s material) WeBWorK 42, 43

More information

Math 140, c Benjamin Aurispa. 2.1 Matrices

Math 140, c Benjamin Aurispa. 2.1 Matrices 2.1 Matrices Reminder: A matrix with m rows and n columns has size m x n. (This is also sometimes referred to as the order of the matrix.) The entry in the ith row and jth column of a matrix A is denoted

More information

Multiplying matrices by diagonal matrices is faster than usual matrix multiplication.

Multiplying matrices by diagonal matrices is faster than usual matrix multiplication. 7-6 Multiplying matrices by diagonal matrices is faster than usual matrix multiplication. The following equations generalize to matrices of any size. Multiplying a matrix from the left by a diagonal matrix

More information

Fall Inverse of a matrix. Institute: UC San Diego. Authors: Alexander Knop

Fall Inverse of a matrix. Institute: UC San Diego. Authors: Alexander Knop Fall 2017 Inverse of a matrix Authors: Alexander Knop Institute: UC San Diego Row-Column Rule If the product AB is defined, then the entry in row i and column j of AB is the sum of the products of corresponding

More information

Matrix Algebra: Definitions and Basic Operations

Matrix Algebra: Definitions and Basic Operations Section 4 Matrix Algebra: Definitions and Basic Operations Definitions Analyzing economic models often involve working with large sets of linear equations. Matrix algebra provides a set of tools for dealing

More information

Review of linear algebra

Review of linear algebra Review of linear algebra 1 Vectors and matrices We will just touch very briefly on certain aspects of linear algebra, most of which should be familiar. Recall that we deal with vectors, i.e. elements of

More information

Basic Concepts in Linear Algebra

Basic Concepts in Linear Algebra Basic Concepts in Linear Algebra Grady B Wright Department of Mathematics Boise State University February 2, 2015 Grady B Wright Linear Algebra Basics February 2, 2015 1 / 39 Numerical Linear Algebra Linear

More information

MAT 2037 LINEAR ALGEBRA I web:

MAT 2037 LINEAR ALGEBRA I web: MAT 237 LINEAR ALGEBRA I 2625 Dokuz Eylül University, Faculty of Science, Department of Mathematics web: Instructor: Engin Mermut http://kisideuedutr/enginmermut/ HOMEWORK 2 MATRIX ALGEBRA Textbook: Linear

More information

Lecture 3 Linear Algebra Background

Lecture 3 Linear Algebra Background Lecture 3 Linear Algebra Background Dan Sheldon September 17, 2012 Motivation Preview of next class: y (1) w 0 + w 1 x (1) 1 + w 2 x (1) 2 +... + w d x (1) d y (2) w 0 + w 1 x (2) 1 + w 2 x (2) 2 +...

More information

1 Matrices and Systems of Linear Equations. a 1n a 2n

1 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 information

POLI270 - Linear Algebra

POLI270 - 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 information

Review of Basic Concepts in Linear Algebra

Review of Basic Concepts in Linear Algebra Review of Basic Concepts in Linear Algebra Grady B Wright Department of Mathematics Boise State University September 7, 2017 Math 565 Linear Algebra Review September 7, 2017 1 / 40 Numerical Linear Algebra

More information

MATH2210 Notebook 2 Spring 2018

MATH2210 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 information

Stage-structured Populations

Stage-structured Populations Department of Biology New Mexico State University Las Cruces, New Mexico 88003 brook@nmsu.edu Fall 2009 Age-Structured Populations All individuals are not equivalent to each other Rates of survivorship

More information

. =. a i1 x 1 + a i2 x 2 + a in x n = b i. a 11 a 12 a 1n a 21 a 22 a 1n. i1 a i2 a in

. =. a i1 x 1 + a i2 x 2 + a in x n = b i. a 11 a 12 a 1n a 21 a 22 a 1n. i1 a i2 a in Vectors and Matrices Continued Remember that our goal is to write a system of algebraic equations as a matrix equation. Suppose we have the n linear algebraic equations a x + a 2 x 2 + a n x n = b a 2

More information

Definition 2.3. We define addition and multiplication of matrices as follows.

Definition 2.3. We define addition and multiplication of matrices as follows. 14 Chapter 2 Matrices In this chapter, we review matrix algebra from Linear Algebra I, consider row and column operations on matrices, and define the rank of a matrix. Along the way prove that the row

More information

Introduction - Motivation. Many phenomena (physical, chemical, biological, etc.) are model by differential equations. f f(x + h) f(x) (x) = lim

Introduction - Motivation. Many phenomena (physical, chemical, biological, etc.) are model by differential equations. f f(x + h) f(x) (x) = lim Introduction - Motivation Many phenomena (physical, chemical, biological, etc.) are model by differential equations. Recall the definition of the derivative of f(x) f f(x + h) f(x) (x) = lim. h 0 h Its

More information

Matrices. 1 a a2 1 b b 2 1 c c π

Matrices. 1 a a2 1 b b 2 1 c c π Matrices 2-3-207 A matrix is a rectangular array of numbers: 2 π 4 37 42 0 3 a a2 b b 2 c c 2 Actually, the entries can be more general than numbers, but you can think of the entries as numbers to start

More information

Two matrices of the same size are added by adding their corresponding entries =.

Two matrices of the same size are added by adding their corresponding entries =. 2 Matrix algebra 2.1 Addition and scalar multiplication Two matrices of the same size are added by adding their corresponding entries. For instance, 1 2 3 2 5 6 3 7 9 +. 4 0 9 4 1 3 0 1 6 Addition of two

More information

7.5 Operations with Matrices. Copyright Cengage Learning. All rights reserved.

7.5 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 information

Chapter 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 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 information

What is the Matrix? Linear control of finite-dimensional spaces. November 28, 2010

What is the Matrix? Linear control of finite-dimensional spaces. November 28, 2010 What is the Matrix? Linear control of finite-dimensional spaces. November 28, 2010 Scott Strong sstrong@mines.edu Colorado School of Mines What is the Matrix? p. 1/20 Overview/Keywords/References Advanced

More information

Section 12.4 Algebra of Matrices

Section 12.4 Algebra of Matrices 244 Section 2.4 Algebra of Matrices Before we can discuss Matrix Algebra, we need to have a clear idea what it means to say that two matrices are equal. Let's start a definition. Equal Matrices Two matrices

More information

Matrix operations Linear Algebra with Computer Science Application

Matrix operations Linear Algebra with Computer Science Application Linear Algebra with Computer Science Application February 14, 2018 1 Matrix operations 11 Matrix operations If A is an m n matrix that is, a matrix with m rows and n columns then the scalar entry in the

More information

Matrix Basic Concepts

Matrix Basic Concepts Matrix Basic Concepts Topics: What is a matrix? Matrix terminology Elements or entries Diagonal entries Address/location of entries Rows and columns Size of a matrix A column matrix; vectors Special types

More information

MTH5112 Linear Algebra I MTH5212 Applied Linear Algebra (2017/2018)

MTH5112 Linear Algebra I MTH5212 Applied Linear Algebra (2017/2018) MTH5112 Linear Algebra I MTH5212 Applied Linear Algebra (2017/2018) COURSEWORK 3 SOLUTIONS Exercise ( ) 1. (a) Write A = (a ij ) n n and B = (b ij ) n n. Since A and B are diagonal, we have a ij = 0 and

More information

Computational Foundations of Cognitive Science. Addition and Scalar Multiplication. Matrix Multiplication

Computational Foundations of Cognitive Science. Addition and Scalar Multiplication. Matrix Multiplication Computational Foundations of Cognitive Science Lecture 1: Algebraic ; Transpose; Inner and Outer Product Frank Keller School of Informatics University of Edinburgh keller@inf.ed.ac.uk February 23, 21 1

More information

Math 4377/6308 Advanced Linear Algebra

Math 4377/6308 Advanced Linear Algebra 1.3 Subspaces Math 4377/6308 Advanced Linear Algebra 1.3 Subspaces Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math4377 Jiwen He, University of Houston

More information

Linear Algebra March 16, 2019

Linear Algebra March 16, 2019 Linear Algebra March 16, 2019 2 Contents 0.1 Notation................................ 4 1 Systems of linear equations, and matrices 5 1.1 Systems of linear equations..................... 5 1.2 Augmented

More information

MATH 323 Linear Algebra Lecture 6: Matrix algebra (continued). Determinants.

MATH 323 Linear Algebra Lecture 6: Matrix algebra (continued). Determinants. MATH 323 Linear Algebra Lecture 6: Matrix algebra (continued). Determinants. Elementary matrices Theorem 1 Any elementary row operation σ on matrices with n rows can be simulated as left multiplication

More information

Knowledge Discovery and Data Mining 1 (VO) ( )

Knowledge Discovery and Data Mining 1 (VO) ( ) Knowledge Discovery and Data Mining 1 (VO) (707.003) Review of Linear Algebra Denis Helic KTI, TU Graz Oct 9, 2014 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, 2014 1 / 74 Big picture: KDDM Probability Theory

More information

Linear Algebra V = T = ( 4 3 ).

Linear Algebra V = T = ( 4 3 ). Linear Algebra Vectors A column vector is a list of numbers stored vertically The dimension of a column vector is the number of values in the vector W is a -dimensional column vector and V is a 5-dimensional

More information

Linear Algebra (Review) Volker Tresp 2018

Linear Algebra (Review) Volker Tresp 2018 Linear Algebra (Review) Volker Tresp 2018 1 Vectors k, M, N are scalars A one-dimensional array c is a column vector. Thus in two dimensions, ( ) c1 c = c 2 c i is the i-th component of c c T = (c 1, c

More information

ICS141: Discrete Mathematics for Computer Science I

ICS141: Discrete Mathematics for Computer Science I ICS4: Discrete Mathematics for Computer Science I Dept. Information & Computer Sci., Jan Stelovsky based on slides by Dr. Baek and Dr. Still Originals by Dr. M. P. Frank and Dr. J.L. Gross Provided by

More information

Lesson U2.1 Study Guide

Lesson U2.1 Study Guide Lesson U2.1 Study Guide Sunday, June 3, 2018 2:05 PM Matrix operations, The Inverse of a Matrix and Matrix Factorization Reading: Lay, Sections 2.1, 2.2, 2.3 and 2.5 (about 24 pages). MyMathLab: Lesson

More information

ORIE 6300 Mathematical Programming I August 25, Recitation 1

ORIE 6300 Mathematical Programming I August 25, Recitation 1 ORIE 6300 Mathematical Programming I August 25, 2016 Lecturer: Calvin Wylie Recitation 1 Scribe: Mateo Díaz 1 Linear Algebra Review 1 1.1 Independence, Spanning, and Dimension Definition 1 A (usually infinite)

More information

We could express the left side as a sum of vectors and obtain the Vector Form of a Linear System: a 12 a x n. a m2

We could express the left side as a sum of vectors and obtain the Vector Form of a Linear System: a 12 a x n. a m2 Week 22 Equations, Matrices and Transformations Coefficient Matrix and Vector Forms of a Linear System Suppose we have a system of m linear equations in n unknowns a 11 x 1 + a 12 x 2 + + a 1n x n b 1

More information

Math Linear Algebra Final Exam Review Sheet

Math 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 information

[ Here 21 is the dot product of (3, 1, 2, 5) with (2, 3, 1, 2), and 31 is the dot product of

[ 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 information