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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 1 Homework, Page 590 Determine the order of the matrix Indicate whether the matrix is square. 1. The matrix has order 2 x 3, and it is not a square matrix.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 2 Homework, Page 590 Determine the order of the matrix Indicate whether the matrix is square. 5. The matrix has order 3 x 1, and it is not a square matrix.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 3 Homework, Page 590 Identify the element specified for the following matrix. 9.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 4 Homework, Page 590 Find (a) A + B, (b) A - B, (c) 3A, and (d) 2A - 3B. 13.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 5 Homework, Page 590 Use the definition of matrix multiplication to find (a) AB and (b) BA. Support your answer with your grapher. 17.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 6 Homework, Page 590 Use the definition of matrix multiplication to find (a) AB and (b) BA. Support your answer with your grapher. 21.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 7 Homework, Page 590 Use the definition of matrix multiplication to find (a) AB and (b) BA. Support your answer with your grapher. 21.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 8 Homework, Page 590 Find (a) AB and (b) BA or state that the product is undefined. 25.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 9 Homework, Page 590 Solve for a and b. 29.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 10 Homework, Page 590 Verify that the matrices are inverses of each other. 33.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 11 Homework, Page 590 Verify that the matrices are inverses of each other. 33.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 12 Homework, Page 590 Find the inverse of the matrix, if it exists. 37.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 13 Homework, Page 590 Evaluate the inverse of the matrix. 41.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 14 Homework, Page 590 45. The matrix below gives the road mileage between (A) Atlanta, (B) Baltimore, (C) Cleveland, and (D) Denver. (a) Explain why the entry in the i th row and j th column is the same as the entry in the j th row and i th column. A matrix with this property is symmetric. The entry in the i th row and j th column is the distance from city i to city j and the entry in the j th row and i th column is also the distance between city i and city j, so the entries must be the same. (b) Why are the entries along the main diagonal all zeroes? The distance from a city to itself is always zero, thus the zero entries.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 15 Homework, Page 590 49. A furniture store has four types of five piece bedroom sets. The price charged for a bedroom set of type j is represented by a 1j. The number of sets of type j sold in one period is represented by b 1j in the matrix The cost to the furniture store for a bedroom set of type j is given by c 1j in the matrix (a) Write a matrix product that gives the total revenue made from the sale of the bedroom sets in one period. (b) Write a matrix expression that gives the profit produced by the sale of the bedroom sets in one period.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 16 Homework, Page 590 Prove that the image of a point under the given transformation of the plane can be obtained by matrix multiplication. 57. A reflection across the y-axis. Even functions are reflections across the y- axis and by definition, an even function is If we represent a point on the Cartesian plane by the matrix [x y], it can be transformed into the point [– x y] by matrix multiplication.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 17 Homework, Page 590 Prove that the image of a point under the given transformation of the plane can be obtained by matrix multiplication. 61. A horizontal stretch or shrink by a factor of c. If a horizontal stretch or shrink is applied to a graph, each point (x, y) on the graph becomes (cx, y). If we represent a point on the Cartesian plane by the matrix [x y], it can be transformed into the point [cx y] by matrix multiplication.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 18 Homework, Page 590 65. Let A be a matrix of order 3 x 2 and B be a matrix of order 2 x 4. which of the following gives the order of the product AB? (a)2 x 2 (b)3 x 4 (c)4 x 3 (d)6 x 8 (e)The product is not defined.

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

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 20 Quick Review

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 21 Quick Review Solutions

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 22 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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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 23 Triangular Form of a System of Equations A system of equations is said to be in triangular form, if it has as many equations as variables and if the equations are arranged in such a manner that the top equation has all variables, the next lacks one variable, the next lacks the first variable and a second and so on. For example,

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 24 Example Solving a System of Equations in Triangular Form by Substitution Solve the system.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 25 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. These operations, when used to reduce a system to triangular form, are called Gaussian elimination.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 26 Example Solving a System of Equations Using Gaussian Elimination Solve the system

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 27 Example Solving a System of Equations Using Gaussian Elimination Solve the system

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 28 Augmented Matrix An augmented matrix is one in which there is one more column than row and where the first columns are the coefficients of a system of equations and the last column contains the constants of the equations. For instance, the system may be represented by the augmented matrix Augmented matrices may be used to record the steps of the Gaussian elimination process.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 29 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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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 30 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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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 31 Example Finding a Row Echelon Form

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 32 Example Solving a System Using Inverse Matrices

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 33 Example Solving a System Using Inverse Matrices Solve the system.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 34 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 row echelon form.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 35 Example Solving a System Using Reduced Row Echelon Form

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 36 Example Solving a Word Problem 74. Stewart’s Metals has three silver alloys on hand. One is 22% silver, one is 30%, and the third is 42%. How many grams of each alloy are required to produce 80 grams of a new alloy that is 34% silver if the amount of the 30% alloy is twice the amount of the 22% alloy used?

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 37 Example Solving a System

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 38 Homework Homework Assignment #11 Read Section 7.4 Page 604, Exercises: 1 – 81 (EOO)

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

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 40 Quick Review

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 41 Quick Review Solutions

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 42 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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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 43 Partial Fraction Decomposition of f(x)/d(x)

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 44 Example Decomposing a Fraction with Distinct Linear Factors

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 45 Example Decomposing a Fraction with Repeated Linear Factors

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 46 Example Decomposing a Fraction with an Irreducible Quadratic Factor

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7- 47 Example Reversing a Decomposed Fraction to Identify the Parent Function

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8.1 Matrices & Systems of Equations

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