Algebra 2 Notes Systems of Equations and Inequalities Unit 03d. Operations with Matrices
|
|
- Daniela Patrick
- 6 years ago
- Views:
Transcription
1 Operations with Matrices Big Idea Organizing data into a matrix can make analysis and interpretation much easier. Operations such as addition, subtraction, and scalar multiplication can be performed on large sets of data in matrix form. To add or subtract two matrices, the matrices must have the same dimensions. Then, calculate the sum or difference of the corresponding entries. The resulting matrix will have the same dimensions as original matrices being combined. Scalar multiplication is done by distributing the scalar to every element within the matrix being multiplied. The result is a product with same dimensions as the original matrix. Objectives: A.CED.3 A. Create equations that describe numbers or relationships - Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Skills Analyze data in matrices Perform algebraic operations with matrices Vocabulary Scalar - A constant. Scalar multiplication - Multiplying any matrix by a scalar. Algebra 2 Unit 03d Operations on Matrices rev Page of 9 4/7/204
2 Examples, Notes, and Exam Questions Matrix Operations Goal: Learn to add and subtract matrices, use scalar multiplication, and solve a matrix equation A matrix is a rectangular arraignment of numbers in rows and columns. For example a matrix A with nmdimensions has n number of rows and m number of columns. The values within the matrix are known as the entries and are indentified by their position in the matrix. Ex : Determine the dimensions of matrix A and identify the position of each entry A Special Types of Matrices 2 A matrix with one row is defined as a row vector. Example: A matrix with one column is defined as a column vector. Example: 5 0 A matrix with the same number of rows as columns is defined as square matrix. Example: A matrix with all zero entries is defined as a zero matrix. Example: Equal Matrices Two matrices are equal if their dimensions are the same and the corresponding entries are equal. 4 2 Example: Matrix A = Matrix B 4 2 A ; B Ex 2: : Determine if the following matrices are equal A 2; B Algebra 2 Unit 03d Operations on Matrices rev Page 2 of 9 4/7/204
3 Adding & Subtracting Matrices WARNING!!! In order to add or subtract the entries of two matrices, they must have the same dimensions. Adding and subtracting are done by adding or subtracting the corresponding entries of each matrix. 7 3 Ex 3: Add Ex 4: Subtract A non-entry real number is defined as a scalar. It is typically a value that is distributed into a matrix with multiplication. Ex 5: Multiply 3Aif 2 0 A 3 Solving Matrix Equations You can use matrix and their equality concepts to solve matrix equation. Ex 6: Solve the matrix for x and y: x y Ex 7: Solve the matrix for x and y: 0 2 x y 2 9 Algebra 2 Unit 03d Operations on Matrices rev Page 3 of 9 4/7/204
4 . To add or subtract to matrices, what must be true? SAMPLE QUESTIONS 2. Perform the indicated operation(s), if possible a b Tell whether the matrices are equal or not equal a. ; Perform the indicated operation, if possible. If not possible, state the reason a b Perform the indicated operation. 0 0 a b. 6. Perform the indicated operation a Solve the matrix equation for x and y. 3x a y 0 Algebra 2 Unit 03d Operations on Matrices rev Page 4 of 9 4/7/204
5 8. Use matrices to organize the following information about car insurance rates. This year: For car, Comprehensive, collision, and basic insurance cost $62.5, $58.29, and $ For 2 cars, Comprehensive, collision, and basic insurance cost $50.32, $984.6, and $ Next year: For car, Comprehensive, collision, and basic insurance cost $66.28, $520.39, and $ For 2 cars, Comprehensive, collision, and basic insurance cost $55.84, $987.82, and $ Algebra 2 Unit 03d Operations on Matrices rev Page 5 of 9 4/7/204
6 Multiplying Matrices Big Idea Matrix multiplication can be performed if the number of columns in the first matrix is equal to the number of rows in the second matrix. This is done by multiplying the elements in the first row by the corresponding elements of the second matrices columns. The resulting products are combined to form a single entry. The matrix dimensions will consist of the number of rows in the first matrix by the number of columns in the second matrix. Objectives: A.CED.3 A. Create equations that describe numbers or relationships - Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Skills Multiply matrices Use the properties of matrix multiplication Vocabulary N/A Algebra 2 Unit 03d Operations on Matrices rev Page 6 of 9 4/7/204
7 Examples, Notes, and Exam Questions Multiplying Matrices Goal: Learn to multiply two matrices Remember: In order to add or subtract two matrices they must have the same Ex 8: Subtract dimension. WARNING!!! To multiply two matrices A and B, the number of columns in A must equal the number of rows in B. Example: if A is a mnmatrix, then B must be an n p, and the product results in a m pmatrix. 2 0 If A and B 3 0, then 45 AB Multiplication Process for Matrices 2 0 Ex 9: : st. Multiply the corresponding entries of row one with those of column one in the second matrix and sum the products into a single solution. 2 nd. The product sum from step one is the first entry of your solution matrix. 3 rd. Repeat steps and 2 until each row of the first matrix has been multiplied by each column of the second matrix. Ex 0: Algebra 2 Unit 03d Operations on Matrices rev Page 7 of 9 4/7/204
8 Ex : Let WARNING!!! Matrix multiplication is NOT commutative. A and B 2, then find AB and BA Ex 2: If A, B,& C , find B( A C) and BA BC. Algebra 2 Unit 03d Operations on Matrices rev Page 8 of 9 4/7/204
9 SAMPLE QUESTIONS. Complete this statement: The product of matrices A and B is defined provided the number of in A is equal to the number of in B. 2. State whether the product of AB is defined. If so, give the dimensions of AB. a. A 33; B Find the product. 2 a Two lacrosse teams submit equipment lists to their sponsors. Women s team: 5 sticks, 5 balls, and 6 uniforms. Men s team: 8 sticks, 22 balls, and 7 uniforms. Each stick costs $55, each ball costs $6, and each uniform costs $35. Use matrix multiplication to find the total cost of the equipment for each team. 5. State whether the product AB is defined. If so, give the dimensions of AB. a. A: 24, B: 4 3 b. A: 24, B: Find the product. If it is not defined, state the reason a Use the given matrices, simplify the expression A ; B ; C ; D 2 4 ; E a. 2AB b. E D E Algebra 2 Unit 03d Operations on Matrices rev Page 9 of 9 4/7/204
10 8. Solve for x and y a x y A Algebra 2 Unit 03d Operations on Matrices rev Page 0 of 9 4/7/204
11 Solving Systems of Equations using Cramer's Rule Big Idea Systems of equations can be written as a matrix. In this form Cramer's Rule can be used to solve the system. Cramer's Rule utilizes the determinant which can be found by calculating the difference of the downward diagonals with upward diagonals. It is important to note that a system as no solution if the determinant in the denominator is zero. Objectives: A.CED.3 A. Create equations that describe numbers or relationships - Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Skills Evaluate determinants Solve system of linear equations by using Cramer's Rule Vocabulary Determinant - A square array of numbers or variables enclosed between two parallel lines. Second-order determinant - The determinant of a 22matrix. Third-order determinant - The determinant of a 33matrix. Diagonal rule - A method for finding the determinant of a 33matrix. Cramer's Rule - A method that uses determinants to solve a system of equations. Coefficient matrix - A matrix that contains only coefficients of a system of equations. Algebra 2 Unit 03d Operations on Matrices rev Page of 9 4/7/204
12 Examples, Notes, and Exam Questions Determinants and Cramer s Rule Goal: Learn to evaluate the determinant of a 22&33matrix, and then use Cramer s Rule to solve a system of equations. Systems of equations can be solved by graphing, substitution, or elimination (combination). All methods define the solution of the system. That is, the input value (x) that produces an equal output value (y) for both functions, and the point where the lines intersect. Solve the following system: 2 x y 3x 2y 23 Determinants and Cramer s Rule provide a matrix approach to solving linear systems. A Determinant of a 22matrix is the difference of the products of the entries on the diagonals. The determinant of a matrix A is denoted by det A or A. a Example: Given A c b d, find A. Solution: det A A ad c b Example: Evaluate the determinant of the two by two matrix: A The determinant of a 33matrix is the difference between the sums of the products of the entries on the diagonals. a b c Example: Given B d e f, finddet B. Solution: g h i det B aei b f g cd h g ec h f a id b Example: Evaluate the determinant of the 33matrix: 4 3 A Determinants can be used to find the area of a triangle whose vertices are points in a coordinate plane. The area of a triangle in a coordinate plane can be found using: Algebra 2 Unit 03d Operations on Matrices rev Page 2 of 9 4/7/204
13 x y Area x y 2 x y , the ensures a positive area. Example: Find the area of a triangle with vertices at 2, 2, 2,4,& 5,. Cramer s Rule is a matrix method for solving linear systems, which utilizes the coefficient matrix and determinants. Determining the coefficient matrix ax by e Given a linear system cx dy f, the coefficient matrix is defined by the coefficients of x and y; a b c d. Example: Determine the coefficient matrix for the linear system 2 x y and then write as a matrix 3x 2y 23 equation. Steps to solve a linear system using Cramer s Rule st. Determine the coefficient matrix Example: 4 x 6 y 4 x5y 4 2 nd. Calculate the determinant of the coefficient matrix found is step. 3 rd. Substitute the constant values in for the x coefficients of the coefficient matrix and calculate this determinant. 4 th. Substitute the constant values in for the y coefficients of the coefficient matrix and calculate the determinant. 5 th. Divide the determinant values found in steps 3 and 4 by the determinant value of the coefficient matrix found in step 2. Note: If the determinant of the coefficient matrix does NOT equal zero, then the system has exactly one solution. Example: Solve 8 x 5 y 2 2x 4y 0 Algebra 2 Unit 03d Operations on Matrices rev Page 3 of 9 4/7/204
14 SAMPLE QUESTIONS. Evaluate the determinant of the matrix. 3 0 a. 2 2 b Evaluate the determinant of the matrix. 3 2 a Use Cramer s rule to solve the linear system. 6x8y 4 a. 4x 5y 4 b. 2 x 7 y 3 3x8y What is the determinant of A. 7 B. 2 C. D ? 5. Cramer s Rule is used to solve the system of equations below. 4x 5 y z 3x 2 y 2z 5 2x 6 y 3z 8 Which determinant represents the denominator for the solution of z? 4 5 E F Algebra 2 Unit 03d Operations on Matrices rev Page 4 of 9 4/7/204
15 G. H Algebra 2 Unit 03d Operations on Matrices rev Page 5 of 9 4/7/204
16 Solving Systems of Equations Using Inverse Matrices (Section 8) Big Idea The identity matrix is a square matrix with diagonal entries of and 0 for all other entries. The product of a matrix and it's inverse is the identity matrix. Thus, an inverse matrix can be used to solve a system of equations written in matrix form. An inverse matrix is found by the product of two factors. The first of which is one of the determinant of the coefficient matrix and the other is the modified coefficient matrix where the entries of the downward diagonals are switched and the upward diagonal entries are negated. Objectives: A.CED.3 A. Create equations that describe numbers or relationships - Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Skills Find the inverse of a 22matrix Write and solve matrix equations for a system of equations Vocabulary Identity matrix - A square matrix that, when multiplied by another matrix, equals that same matrix. If A is any nnmatrix and I is the nnidentity matrix, then AI Aand I A A. Square matrix - A matrix with the same number of rows as columns. Inverse matrix - Two nnmatrices are inverses of each other their product is the identity matrix. Matrix equation - A matrix form used to represent a system of equations. Variable matrix - A matrix that only contains the variables of a system of equations. Constant matrix - A matrix that only contains the constants of a system of equations. Algebra 2 Unit 03d Operations on Matrices rev Page 6 of 9 4/7/204
17 Examples, Notes, and Exam Questions Goal: Learn to identify and use inverse matrices to solve linear systems of equations. Remember: To find the determinant of a 22matrix we take the difference of the products of the diagonal entries. 2 4 Example: Evaluate the determinant of A 3 Additionally, we should recall that the Inverse property states; for any real number a 0, a a where is the multiplicative identity. We use this concept of Multiplicative Inverse to solve problems such as 4x 2 for x, by multiplying both sides of the equation by the inverse of 4. Two nnmatrices are inverses of each other if their product equals the nnidentity matrix. An n n identity matrix has ones on the downward diagonal and zeros for every other entry. Example: 22Identity Matrix: 33Identity Matrix: The inverse matrix can be used to solve matrix equations and is denoted by A. The inverse of a matrix a b d b d b A c d is; A, ad bc 0 A c a ad bc c a Steps to find the inverse of a matrix st. Find the determinant of A. Example: Find the inverse of A nd. Put matrix A in inverse form, that is switch the a and d entries and negate the b and c entries. 3 rd. Distribute the ratio found in step into the matrix formed in step 2. Steps to solve a matrix equation with the inverse matrix st. Determine the matrix equation AX B Example: X nd. Find the inverse A. Algebra 2 Unit 03d Operations on Matrices rev Page 7 of 9 4/7/204
18 3 rd. Multiply both sides of the equation by A. Be sure to multiply to the left of each matrix. That is, A AX A B Inverse matrices can be used to solve a system of equations Steps to solving a linear system of equations with inverse matrices 3x 4y 5 st. Write the system of equations as a Example: Solve 2x y 0 matrix equation. AX B, where A is the coefficient matrix and B is the constant matrix 2 nd. Find the inverse A of the coefficient matrix. 3 rd. Multiply both sides of the matrix equation by A. Algebra 2 Unit 03d Operations on Matrices rev Page 8 of 9 4/7/204
19 SAMPLE QUESTIONS. Find the inverse of the matrix. 5 4 a. 4 4 b Solve the matrix equation a. X b X Write the linear system as a matrix equation. x3z 6 2x 3y z 3x y 2z 3 4. Use an inverse matrix to solve the linear system. x 2y 3 2x8y 5. Write the linear system as a matrix equation. 2x 5y 3x 7y5 6. Use an inverse matrix to solve the linear system. 5x 7y 9 2x3y3 7. Solve the matrix equation X QOD: What is the special relationship between a matrix and its inverse? Algebra 2 Unit 03d Operations on Matrices rev Page 9 of 9 4/7/204
Algebra 2 Notes Systems of Equations and Inequalities Unit 03c. System of Equations in Three Variables
System of Equations in Three Variables Big Idea A system of equations in three variables consists of multiple planes graphed on the same coordinate plane. The solutions to these systems consists of a single
More informationAlgebra II Notes Unit Four: Matrices and Determinants
Syllabus Objectives: 4. The student will organize data using matrices. 4.2 The student will simplify matrix expressions using the properties of matrices. Matrix: a rectangular arrangement of numbers (called
More informationH.Alg 2 Notes: Day1: Solving Systems of Equations (Sections ) Activity: Text p. 116
H.Alg 2 Notes: Day: Solving Systems of Equations (Sections 3.-3.3) Activity: Text p. 6 Systems of Equations: A set of or more equations using the same. The graph of each equation is a line. Solutions of
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 informationUnit 1 Matrices Notes Packet Period: Matrices
Algebra 2/Trig Unit 1 Matrices Notes Packet Name: Period: # Matrices (1) Page 203 204 #11 35 Odd (2) Page 203 204 #12 36 Even (3) Page 211 212 #4 6, 17 33 Odd (4) Page 211 212 #12 34 Even (5) Page 218
More informationSection 5.5: Matrices and Matrix Operations
Section 5.5 Matrices and Matrix Operations 359 Section 5.5: Matrices and Matrix Operations Two club soccer teams, the Wildcats and the Mud Cats, are hoping to obtain new equipment for an upcoming season.
More informationHonors Advanced Mathematics Determinants page 1
Determinants page 1 Determinants For every square matrix A, there is a number called the determinant of the matrix, denoted as det(a) or A. Sometimes the bars are written just around the numbers of the
More informationCHAPTER 8: Matrices and Determinants
(Exercises for Chapter 8: Matrices and Determinants) E.8.1 CHAPTER 8: Matrices and Determinants (A) means refer to Part A, (B) means refer to Part B, etc. Most of these exercises can be done without a
More informationSeptember 23, Chp 3.notebook. 3Linear Systems. and Matrices. 3.1 Solve Linear Systems by Graphing
3Linear Systems and Matrices 3.1 Solve Linear Systems by Graphing 1 Find the solution of the systems by looking at the graphs 2 Decide whether the ordered pair is a solution of the system of linear equations:
More information9 Appendix. Determinants and Cramer s formula
LINEAR ALGEBRA: THEORY Version: August 12, 2000 133 9 Appendix Determinants and Cramer s formula Here we the definition of the determinant in the general case and summarize some features Then we show how
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 informationMatrices. 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 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 informationMATRICES. 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 information7.1 Solving Systems of Equations
Date: Precalculus Notes: Unit 7 Systems of Equations and Matrices 7.1 Solving Systems of Equations Syllabus Objectives: 8.1 The student will solve a given system of equations or system of inequalities.
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 informationBasics. A VECTOR is a quantity with a specified magnitude and direction. A MATRIX is a rectangular array of quantities
Some Linear Algebra Basics A VECTOR is a quantity with a specified magnitude and direction Vectors can exist in multidimensional space, with each element of the vector representing a quantity in a different
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 information8.4. Systems of Equations in Three Variables. Identifying Solutions 2/20/2018. Example. Identifying Solutions. Solving Systems in Three Variables
8.4 Systems of Equations in Three Variables Copyright 2010 Pearson Education, Inc. Publishing as Pearson Addison- Wesley Identifying Solutions Solving Systems in Three Variables Dependency, Inconsistency,
More informationMAC Learning Objectives. Learning Objectives (Cont.) Module 10 System of Equations and Inequalities II
MAC 1140 Module 10 System of Equations and Inequalities II Learning Objectives Upon completing this module, you should be able to 1. represent systems of linear equations with matrices. 2. transform a
More informationMatrix Algebra. Learning Objectives. Size of Matrix
Matrix Algebra 1 Learning Objectives 1. Find the sum and difference of two matrices 2. Find scalar multiples of a matrix 3. Find the product of two matrices 4. Find the inverse of a matrix 5. Solve a system
More informationMTH 306 Spring Term 2007
MTH 306 Spring Term 2007 Lesson 3 John Lee Oregon State University (Oregon State University) 1 / 27 Lesson 3 Goals: Be able to solve 2 2 and 3 3 linear systems by systematic elimination of unknowns without
More informationCollege Algebra. Chapter 6. Mary Stangler Center for Academic Success
College Algebra Chapter 6 Note: This review is composed of questions similar to those in the chapter review at the end of chapter 6. This review is meant to highlight basic concepts from chapter 6. It
More informationUNIT 1 DETERMINANTS 1.0 INTRODUCTION 1.1 OBJECTIVES. Structure
UNIT 1 DETERMINANTS Determinants Structure 1.0 Introduction 1.1 Objectives 1.2 Determinants of Order 2 and 3 1.3 Determinants of Order 3 1.4 Properties of Determinants 1.5 Application of Determinants 1.6
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 informationCHAPTER 7: Systems and Inequalities
(Exercises for Chapter 7: Systems and Inequalities) E.7.1 CHAPTER 7: Systems and Inequalities (A) means refer to Part A, (B) means refer to Part B, etc. (Calculator) means use a calculator. Otherwise,
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 information1 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 informationDefinition of Equality of Matrices. Example 1: Equality of Matrices. Consider the four matrices
IT 131: Mathematics for Science Lecture Notes 3 Source: Larson, Edwards, Falvo (2009): Elementary Linear Algebra, Sixth Edition. Matrices 2.1 Operations with Matrices This section and the next introduce
More information4-1 Matrices and Data
4-1 Matrices and Data Warm Up Lesson Presentation Lesson Quiz 2 The table shows the top scores for girls in barrel racing at the 2004 National High School Rodeo finals. The data can be presented in a table
More informationMatrices 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 informationMatrix Operations (Adding, Subtracting, and Multiplying) Ch. 3.5, 6 Finding the Determinant and Inverse of a Matrix Ch. 3.7, 8
Matrix Operations (Adding, Subtracting, and Multiplying) Ch. 3.5, 6 Finding the Determinant and Inverse of a Matrix Ch. 3.7, 8 Objectives: Add, subtract, and multiply matrices. Find the determinant and
More informationMAC1105-College Algebra. Chapter 5-Systems of Equations & Matrices
MAC05-College Algebra Chapter 5-Systems of Equations & Matrices 5. Systems of Equations in Two Variables Solving Systems of Two Linear Equations/ Two-Variable Linear Equations A system of equations is
More informationChapter 2 Linear Equations and Inequalities in One Variable
Chapter 2 Linear Equations and Inequalities in One Variable Section 2.1: Linear Equations in One Variable Section 2.3: Solving Formulas Section 2.5: Linear Inequalities in One Variable Section 2.6: Compound
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 informationCalculus II - Basic Matrix Operations
Calculus II - Basic Matrix Operations Ryan C Daileda Terminology A matrix is a rectangular array of numbers, for example 7,, 7 7 9, or / / /4 / / /4 / / /4 / /6 The numbers in any matrix are called its
More informationAlgebra I Teacher Notes Expressions, Equations, and Formulas Unit 01 Review
Big Ideas Write and evaluate algebraic expressions Use expressions to write equations and inequalities Solve equations Represent functions as verbal rules, equations, tables and graphs Review concepts
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 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 informationAlgebra II Notes Quadratic Functions Unit Complex Numbers. Math Background
Complex Numbers Math Background Previously, you Studied the real number system and its sets of numbers Applied the commutative, associative and distributive properties to real numbers Used the order of
More information3. Vector spaces 3.1 Linear dependence and independence 3.2 Basis and dimension. 5. Extreme points and basic feasible solutions
A. LINEAR ALGEBRA. CONVEX SETS 1. Matrices and vectors 1.1 Matrix operations 1.2 The rank of a matrix 2. Systems of linear equations 2.1 Basic solutions 3. Vector spaces 3.1 Linear dependence and independence
More information2.1 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 informationThe word Matrices is the plural of the word Matrix. A matrix is a rectangular arrangement (or array) of numbers called elements.
Numeracy Matrices Definition The word Matrices is the plural of the word Matrix A matrix is a rectangular arrangement (or array) of numbers called elements A x 3 matrix can be represented as below Matrix
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 informationPhys 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 informationn 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 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 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 informationTOPIC III LINEAR ALGEBRA
[1] Linear Equations TOPIC III LINEAR ALGEBRA (1) Case of Two Endogenous Variables 1) Linear vs. Nonlinear Equations Linear equation: ax + by = c, where a, b and c are constants. 2 Nonlinear equation:
More informationDependent ( ) Independent (1 or Ø) These lines coincide so they are a.k.a coincident.
Notes #3- Date: 7.1 Solving Systems of Two Equations (568) The solution to a system of linear equations is the ordered pair (x, y) where the lines intersect! A solution can be substituted into both equations
More informationMATRIX DETERMINANTS. 1 Reminder Definition and components of a matrix
MATRIX DETERMINANTS Summary Uses... 1 1 Reminder Definition and components of a matrix... 1 2 The matrix determinant... 2 3 Calculation of the determinant for a matrix... 2 4 Exercise... 3 5 Definition
More informationMath 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 informationMatrix 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 informationQuadratic Functions. Key Terms. Slide 1 / 200. Slide 2 / 200. Slide 3 / 200. Table of Contents
Slide 1 / 200 Quadratic Functions Table of Contents Key Terms Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic Equations
More informationQuadratic Functions. Key Terms. Slide 2 / 200. Slide 1 / 200. Slide 3 / 200. Slide 4 / 200. Slide 6 / 200. Slide 5 / 200.
Slide 1 / 200 Quadratic Functions Slide 2 / 200 Table of Contents Key Terms Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic
More informationSlide 1 / 200. Quadratic Functions
Slide 1 / 200 Quadratic Functions Key Terms Slide 2 / 200 Table of Contents Identify Quadratic Functions Explain Characteristics of Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic
More informationSections 6.1 and 6.2: Systems of Linear Equations
What is a linear equation? Sections 6.1 and 6.2: Systems of Linear Equations We are now going to discuss solving systems of two or more linear equations with two variables. Recall that solving an equation
More informationConceptual Questions for Review
Conceptual Questions for Review Chapter 1 1.1 Which vectors are linear combinations of v = (3, 1) and w = (4, 3)? 1.2 Compare the dot product of v = (3, 1) and w = (4, 3) to the product of their lengths.
More informationOrder of Operations. Real numbers
Order of Operations When simplifying algebraic expressions we use the following order: 1. Perform operations within a parenthesis. 2. Evaluate exponents. 3. Multiply and divide from left to right. 4. Add
More informationCCSS Math- Algebra. Domain: Algebra Seeing Structure in Expressions A-SSE. Pacing Guide. Standard: Interpret the structure of expressions.
1 Domain: Algebra Seeing Structure in Expressions A-SSE Standard: Interpret the structure of expressions. H.S. A-SSE.1a. Interpret expressions that represent a quantity in terms of its context. Content:
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 informationDefinition. A matrix is a rectangular array of numbers enclosed by brackets (plural: matrices).
Matrices (general theory). Definition. A matrix is a rectangular array of numbers enclosed by brackets (plural: matrices). Examples. 1 2 1 1 0 2 A= 0 0 7 B= 0 1 3 4 5 0 Terminology and Notations. Each
More informationSystems of Equations and Inequalities
7 Systems of Equations and Inequalities CHAPTER OUTLINE Introduction Figure 1 Enigma machines like this one, once owned by Italian dictator Benito Mussolini, were used by government and military officials
More informationChapter 2. Matrix Arithmetic. Chapter 2
Matrix Arithmetic Matrix Addition and Subtraction Addition and subtraction act element-wise on matrices. In order for the addition/subtraction (A B) to be possible, the two matrices A and B must have the
More information5 Systems of Equations
Systems of Equations Concepts: Solutions to Systems of Equations-Graphically and Algebraically Solving Systems - Substitution Method Solving Systems - Elimination Method Using -Dimensional Graphs to Approximate
More informationSOLUTIONS FOR PROBLEMS 1-30
. Answer: 5 Evaluate x x + 9 for x SOLUTIONS FOR PROBLEMS - 0 When substituting x in x be sure to do the exponent before the multiplication by to get (). + 9 5 + When multiplying ( ) so that ( 7) ( ).
More informationM. Matrices and Linear Algebra
M. Matrices and Linear Algebra. Matrix algebra. In section D we calculated the determinants of square arrays of numbers. Such arrays are important in mathematics and its applications; they are called matrices.
More informationBasic 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 informationMatrix 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 informationMatrices. 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 information1 4 3 A Scalar Multiplication
1 Matrices A matrix is a rectangular array of variables or constants in horizontal rows and vertical columns, usually enclosed in brackets. In a matrix, the numbers or data are organized so that each position
More informationReview 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 informationMatrix Operations and Equations
C H A P T ER Matrix Operations and Equations 200 Carnegie Learning, Inc. Shoe stores stock various sizes and widths of each style to accommodate buyers with different shaped feet. You will use matrix operations
More informationElementary 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 informationMath Studio College Algebra
Math 100 - Studio College Algebra Rekha Natarajan Kansas State University November 19, 2014 Systems of Equations Systems of Equations A system of equations consists of Systems of Equations A system of
More informationMATRICES The numbers or letters in any given matrix are called its entries or elements
MATRICES A matrix is defined as a rectangular array of numbers. Examples are: 1 2 4 a b 1 4 5 A : B : C 0 1 3 c b 1 6 2 2 5 8 The numbers or letters in any given matrix are called its entries or elements
More informationcourses involve systems of equations in one way or another.
Another Tool in the Toolbox Solving Matrix Equations.4 Learning Goals In this lesson you will: Determine the inverse of a matrix. Use matrices to solve systems of equations. Key Terms multiplicative identity
More informationMath 343 Midterm I Fall 2006 sections 002 and 003 Instructor: Scott Glasgow
Math 343 Midterm I Fall 006 sections 00 003 Instructor: Scott Glasgow 1 Assuming A B are invertible matrices of the same size, prove that ( ) 1 1 AB B A 1 = (11) B A 1 1 is the inverse of AB if only if
More informationFinite Math - J-term Section Systems of Linear Equations in Two Variables Example 1. Solve the system
Finite Math - J-term 07 Lecture Notes - //07 Homework Section 4. - 9, 0, 5, 6, 9, 0,, 4, 6, 0, 50, 5, 54, 55, 56, 6, 65 Section 4. - Systems of Linear Equations in Two Variables Example. Solve the system
More informationMath 313 (Linear Algebra) Exam 2 - Practice Exam
Name: Student ID: Section: Instructor: Math 313 (Linear Algebra) Exam 2 - Practice Exam Instructions: For questions which require a written answer, show all your work. Full credit will be given only if
More informationLS.1 Review of Linear Algebra
LS. LINEAR SYSTEMS LS.1 Review of Linear Algebra In these notes, we will investigate a way of handling a linear system of ODE s directly, instead of using elimination to reduce it to a single higher-order
More informationLinear Algebra Tutorial for Math3315/CSE3365 Daniel R. Reynolds
Linear Algebra Tutorial for Math3315/CSE3365 Daniel R. Reynolds These notes are meant to provide a brief introduction to the topics from Linear Algebra that will be useful in Math3315/CSE3365, Introduction
More informationLinear 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 informationSection 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 informationIntroduction to Vectors Pg. 279 # 1 6, 8, 9, 10 OR WS 1.1 Sept. 7. Vector Addition Pg. 290 # 3, 4, 6, 7, OR WS 1.2 Sept. 8
UNIT 1 INTRODUCTION TO VECTORS Lesson TOPIC Suggested Work Sept. 5 1.0 Review of Pre-requisite Skills Pg. 273 # 1 9 OR WS 1.0 Fill in Info sheet and get permission sheet signed. Bring in $3 for lesson
More informationSystem of Linear Equations. Slide for MA1203 Business Mathematics II Week 1 & 2
System of Linear Equations Slide for MA1203 Business Mathematics II Week 1 & 2 Function A manufacturer would like to know how his company s profit is related to its production level. How does one quantity
More informationAlgebra II Vocabulary Alphabetical Listing. Absolute Maximum: The highest point over the entire domain of a function or relation.
Algebra II Vocabulary Alphabetical Listing Absolute Maximum: The highest point over the entire domain of a function or relation. Absolute Minimum: The lowest point over the entire domain of a function
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 informationMath 346 Notes on Linear Algebra
Math 346 Notes on Linear Algebra Ethan Akin Mathematics Department Fall, 2014 1 Vector Spaces Anton Chapter 4, Section 4.1 You should recall the definition of a vector as an object with magnitude and direction
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 informationCourse Number 432/433 Title Algebra II (A & B) H Grade # of Days 120
Whitman-Hanson Regional High School provides all students with a high- quality education in order to develop reflective, concerned citizens and contributing members of the global community. Course Number
More informationAlgebra II Notes Matrices and Determinants Unit 04
Matrix Addition Big Idea: Matrices are used to organize, display, interpret, and analyze data. Algebraic operations are performed with matrices to solve real world problems. Matrices can be added and subtracted
More informationLinear 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 informationA2H Assignment #8 Cramer s Rule Unit 2: Matrices and Systems. DUE Date: Friday 12/2 as a Packet. 3x 2y = 10 5x + 3y = 4. determinant.
A2H Assignment #8 Cramer s Rule Unit 2: Matrices and Systems Name: DUE Date: Friday 12/2 as a Packet What is the Cramer s Rule used for? à Another method to solve systems that uses matrices and determinants.
More informationReview 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 informationSection 5.3 Systems of Linear Equations: Determinants
Section 5. Systems of Linear Equations: Determinants In this section, we will explore another technique for solving systems called Cramer's Rule. Cramer's rule can only be used if the number of equations
More information9.1 - Systems of Linear Equations: Two Variables
9.1 - Systems of Linear Equations: Two Variables Recall that a system of equations consists of two or more equations each with two or more variables. A solution to a system in two variables is an ordered
More informationMATCHING. Match the correct vocabulary word with its definition
Name Algebra I Block UNIT 2 STUDY GUIDE Ms. Metzger MATCHING. Match the correct vocabulary word with its definition 1. Whole Numbers 2. Integers A. A value for a variable that makes an equation true B.
More informationReference Material /Formulas for Pre-Calculus CP/ H Summer Packet
Reference Material /Formulas for Pre-Calculus CP/ H Summer Packet Week # 1 Order of Operations Step 1 Evaluate expressions inside grouping symbols. Order of Step 2 Evaluate all powers. Operations Step
More information