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Chapter 7

Chapter 7. Systems and Matrices. 7.1. Solving Systems of Two Equations. Quick Review. Quick Review Solutions. What you’ll learn about. The Method of Substitution Solving Systems Graphically The Method of Elimination Applications … and why

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Chapter 7

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  1. Chapter 7 Systems and Matrices

  2. 7.1 Solving Systems of Two Equations

  3. Quick Review

  4. Quick Review Solutions

  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.

  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.

  7. Example Using the Substitution Method

  8. Example Using the Substitution Method

  9. Example Solving a Nonlinear System Algebraically

  10. Example Solving a Nonlinear System Algebraically

  11. Example Using the Elimination Method

  12. Example Using the Elimination Method

  13. Example Finding No Solution

  14. Example Finding No Solution

  15. Example Finding Infinitely Many Solutions

  16. Example Finding Infinitely Many Solutions

  17. 7.2 Matrix Algebra

  18. Quick Review

  19. Quick Review Solutions

  20. 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.

  21. Matrix

  22. Matrix Vocabulary Each element, or entry, aij, of the matrix uses double subscript notation. The row subscript is the first subscript i, and the column subscript is j. The element aij is the ith row and the jth column. In general, the orderof anm × n matrix is m×n.

  23. Example Determining the Order of a Matrix

  24. Example Determining the Order of a Matrix

  25. Matrix Addition and Matrix Subtraction

  26. Example Matrix Addition

  27. Example Matrix Addition

  28. Example Using Scalar Multiplication

  29. Example Using Scalar Multiplication

  30. The Zero Matrix

  31. Additive Inverse

  32. Matrix Multiplication

  33. Example Matrix Multiplication

  34. Example Matrix Multiplication

  35. Identity Matrix

  36. Inverse of a Square Matrix

  37. Inverse of a 2 × 2 Matrix

  38. Determinant of a Square Matrix

  39. Inverses of n× n Matrices An n× n matrix A has an inverse if and only if det A≠ 0.

  40. Example Finding Inverse Matrices

  41. Example Finding Inverse Matrices

  42. 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·In = In·A = A 4. Inverse property Addition: A + (-A) = 0 Multiplication: AA-1 = A-1A = In |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

  43. 7.3 Multivariate Linear Systems and Row Operations

  44. Quick Review

  45. Quick Review Solutions

  46. 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.

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

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

  49. Elementary Row Operations on a Matrix A combination of the following operations will transform a matrix to row echelon form. • Interchange any two rows. • Multiply all elements of a row by a nonzero real number. • Add a multiple of one row to any other row.

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