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Slide 7- 1

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Chapter 7 Systems and Matrices

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7.1 Solving Systems of Two Equations

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Slide 7- 4 Quick Review

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Slide 7- 5 What you’ll learn about The Method of Substitution Solving Systems Graphically The Method of Elimination Applications … and why Many applications in business and science can be modeled using systems of equations.

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Slide 7- 6 Solution of a System A solution of a system of two equations in two variables is an ordered pair of real numbers that is a solution of each equation.

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Slide 7- 7 Example Using the Substitution Method

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Slide 7- 8 Example Using the Substitution Method

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Slide 7- 9 Example Solving a Nonlinear System Algebraically

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Slide 7- 10 Example Using the Elimination Method

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Slide 7- 11 Example Finding No Solution

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Slide 7- 12 Example Finding Infinitely Many Solutions

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7.2 Matrix Algebra

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Slide 7- 14 Quick Review

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Slide 7- 15 What you’ll learn about Matrices Matrix Addition and Subtraction Matrix Multiplication Identity and Inverse Matrices Determinant of a Square Matrix Applications … and why Matrix algebra provides a powerful technique to manipulate large data sets and solve the related problems that are modeled by the matrices.

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Slide 7- 16 Matrix

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Slide 7- 17 Matrix Vocabulary Each element, or entry, a ij, of the matrix uses double subscript notation. The row subscript is the first subscript i, and the column subscript is j. The element a ij is the ith row and the jth column. In general, the order of an m × n matrix is m×n.

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Slide 7- 18 Example Determining the Order of a Matrix

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Slide 7- 19 Matrix Addition and Matrix Subtraction

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Adding Matrices

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Slide 7- 21 Example Matrix Addition

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Slide 7- 22 Example Using Scalar Multiplication

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Slide 7- 23 The Zero Matrix

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Slide 7- 24 Additive Inverse

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Slide 7- 25 Matrix Multiplication

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Let A denote an m by r matrix and let B denote an r by n matrix. The product AB is defined as the m by n matrix whose entry in row i, column j is the product of the ith row of A and the jth column of B. Multiplying Matrices Note: If we multiply a matrix by a constant, this is equivalent to multiplying each term in the matrix by the constant.

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Slide 7- 27 Example Matrix Multiplication

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Slide 7- 28 Example Matrix Multiplication

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Example: Find the product AB

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The Identity Matrix

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Slide 7- 31 Identity Matrix

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Slide 7- 32 Inverse of a Square Matrix

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Slide 7- 33 Inverse of a 2 × 2 Matrix

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Slide 7- 34 Determinant of a Square Matrix

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Slide 7- 35 Inverses of n × n Matrices An n × n matrix A has an inverse if and only if det A ≠ 0.

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Slide 7- 36 Example Finding Inverse Matrices

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Slide 7- 37 Properties of Matrices Let A, B, and C be matrices whose orders are such that the following sums, differences, and products are defined. 1. Community property Addition: A + B = B + A Multiplication: Does not hold in general 2. Associative property Addition: (A + B) + C = A + (B + C) Multiplication: (AB)C = A(BC) 3. Identity property Addition: A + 0 = A Multiplication: A·I n = I n ·A = A 4. Inverse property Addition: A + (-A) = 0 Multiplication: AA -1 = A -1 A = I n |A|≠0 5. Distributive property Multiplication over addition: A(B + C) = AB + AC (A + B)C = AC + BC Multiplication over subtraction: A(B - C) = AB - AC (A - B)C = AC - BC

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Slide 7- 38 Matrices and Transformations P(x,y) P’(x’,y’)

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Slide 7- 39 Matrices and Transformations Rotation through an angle The rotation through an angle maps each point P(x,y) in the rectangular coordinate plane to the point P’(x’,y’). where

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Slide 7- 40 Matrices and Transformations or

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Slide 7- 41 Matrices and Transformations Find the rotation matrix about the origin whose angle is /3.

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Slide 7- 42 Matrices and Transformations Where does the point (4,-2) move?

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7.3 Multivariate Linear Systems and Row Operations

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Slide 7- 44 Quick Review

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Slide 7- 45 What you’ll learn about Triangular Forms for Linear Systems Gaussian Elimination Elementary Row Operations and Row Echelon Form Reduced Row Echelon Form Solving Systems with Inverse Matrices Applications … and why Many applications in business and science are modeled by systems of linear equations in three or more variables.

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Slide 7- 46 Equivalent Systems of Linear Equations The following operations produce an equivalent system of linear equations. 1. Interchange any two equations of the system. 2. Multiply (or divide) one of the equations by any nonzero real number. 3. Add a multiple of one equation to any other equation in the system.

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Slide 7- 47 Row Echelon Form of a Matrix A matrix is in row echelon form if the following conditions are satisfied. 1. Rows consisting entirely of 0’s (if there are any) occur at the bottom of the matrix. 2. The first entry in any row with nonzero entries is 1. 3. The column subscript of the leading 1 entries increases as the row subscript increases.

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Slide 7- 48 Elementary Row Operations on a Matrix A combination of the following operations will transform a matrix to row echelon form. 1. Interchange any two rows. 2. Multiply all elements of a row by a nonzero real number. 3. Add a multiple of one row to any other row.

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Slide 7- 49 Example Finding a Row Echelon Form

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Slide 7- 50 Example Finding a Row Echelon Form

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Slide 7- 51 Reduced Row Echelon Form If we continue to apply elementary row operations to a row echelon form of a matrix, we can obtain a matrix in which every column that has a leading 1 has 0’s elsewhere. This is the reduced echelon form.

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Slide 7- 52 Example Solving a System Using Inverse Matrices

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Slide 7- 53 Example Solving a System Using Inverse Matrices

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Slide 7- 54 Multivariate Linear Systems and Row Operations Page 602

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Slide 7- 55 Multivariate Linear Systems and Row Operations Page 602

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Slide 7- 56 Multivariate Linear Systems and Row Operations

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7.4 Partial Fractions

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Slide 7- 58 Quick Review

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Slide 7- 59 What you’ll learn about Partial Fraction Decomposition Denominators with Linear Factors Denominators with Irreducible Quadratic Factors Applications … and why Partial fraction decompositions are used in calculus in integration and can be used to guide the sketch of the graph of a rational function.

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Slide 7- 60 Partial Fraction Decomposition of f(x)/d(x)

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Slide 7- 61 Example Decomposing a Fraction with Distinct Linear Factors

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Slide 7- 62 Example Decomposing a Fraction with Distinct Linear Factors

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Slide 7- 63 Example Decomposing a Fraction with an Irreducible Quadratic Factor

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Slide 7- 64 Example Decomposing a Fraction with an Irreducible Quadratic Factor

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7.5 Systems of Inequalities in Two Variables

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Slide 7- 66 Quick Review Solutions

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Slide 7- 67 What you’ll learn about Graph of an Inequality Systems of Inequalities Linear Programming … and why Linear programming is used in business and industry to maximize profits, minimize costs, and to help management make decisions.

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Slide 7- 68 Steps for Drawing the Graph of an Inequality in Two Variables 1. Draw the graph of the equation obtained by replacing the inequality sign by an equal sign. Use a dashed line if the inequality is. Use a solid line if the inequality is ≤ or ≥. 2. Check a point in each of the two regions of the plane determined by the graph of the equation. If the point satisfies the inequality, then shade the region containing the point.

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Slide 7- 69 Example Graphing a Linear Inequality

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Slide 7- 70 Example Graphing a Linear Inequality

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Slide 7- 71 Example Solving a System of Inequalities Graphically

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Slide 7- 72 Example Solving a System of Inequalities Graphically

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Slide 7- 73 Chapter Test

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Slide 7- 74 Chapter Test

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Slide 7- 75 Chapter Test

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Slide 7- 76 Chapter Test

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Slide 7- 77 Chapter Test Solutions

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Slide 7- 78 Chapter Test Solutions

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Slide 7- 79 Chapter Test Solutions

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Slide 7- 80 Chapter Test Solutions

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