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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 1

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 7 Systems and Matrices

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 7.1 Solving Systems of Two Equations

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 4 Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 5 Quick Review Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 6 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.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 7 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.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 8 Example Using the Substitution Method

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 9 Example Using the Substitution Method

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Solving a Nonlinear System Algebraically

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Solving a Nonlinear System Algebraically

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Using the Elimination Method

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Using the Elimination Method

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding No Solution

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding No Solution

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding Infinitely Many Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding Infinitely Many Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 7.2 Matrix Algebra

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Quick Review Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Matrix

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Determining the Order of a Matrix

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Determining the Order of a Matrix

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Matrix Addition and Matrix Subtraction

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Matrix Addition

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Matrix Addition

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Using Scalar Multiplication

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Using Scalar Multiplication

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide The Zero Matrix

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Additive Inverse

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Matrix Multiplication

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Matrix Multiplication

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Matrix Multiplication

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Identity Matrix

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Inverse of a Square Matrix

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Inverse of a 2 × 2 Matrix

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Determinant of a Square Matrix

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Inverses of n × n Matrices An n × n matrix A has an inverse if and only if det A ≠ 0.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding Inverse Matrices

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding Inverse Matrices

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 7.3 Multivariate Linear Systems and Row Operations

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Quick Review Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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 The column subscript of the leading 1 entries increases as the row subscript increases.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding a Row Echelon Form

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding a Row Echelon Form

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Solving a System Using Inverse Matrices

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Solving a System Using Inverse Matrices

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 7.4 Partial Fractions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Quick Review Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Partial Fraction Decomposition of f(x)/d(x)

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Decomposing a Fraction with Distinct Linear Factors

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Decomposing a Fraction with Distinct Linear Factors

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Decomposing a Fraction with an Irreducible Quadratic Factor

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Decomposing a Fraction with an Irreducible Quadratic Factor

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 7.5 Systems of Inequalities in Two Variables

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Quick Review Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Graphing a Linear Inequality

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Graphing a Linear Inequality

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Solving a System of Inequalities Graphically

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Solving a System of Inequalities Graphically

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Chapter Test

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Chapter Test

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Chapter Test

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Chapter Test

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Chapter Test Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Chapter Test Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Chapter Test Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Chapter Test Solutions