1. Systems of Linear Equations and Matrices
Chapter 2
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2. Systems of Linear Equations and Matrices
2.2 Solution of Linear Systems by the Gauss-Jordan Method
2.3 Addition and Subtraction of Matrices
2.4 Multiplication of Matrices
2.5 Matrix Inverses
2.6 Input-Output Models
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3. Systems of Linear Equations and Matrices
2.2 Solution of Linear Systems by the Gauss-Jordan Method
Working with some basic Matrix Algebra
2.3 Addition and Subtraction of Matrices
2.4 Multiplication of Matrices
2.5 Matrix Inverses
Solving a system of linear equations using matrix inverses (with Microsoft Excel).
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4. Matrix Algebra Matrix – A rectangular array of numbers.
is a m  n matrix (m rows, n columns), where the entry in the ith row and jth column is aij.
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5. Matrix Algebra Matrices are often named with capital letters (M).
Matrices are classified by size (# of rows  # of columns).
M is a 2 x 3 matrix
Row matrix (row vector) – a matrix containing only 1 row.
Column matrix (column vector) – only 1 column.
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6. Matrix Algebra Matrix Equality
Two matrices are equal if they are the same size and if each pair of corresponding elements is equal.
could be true
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7. 2.3 Addition and Subtraction of Matrices
Adding Matrices
The sum of two m  n matrices X and Y is the m  n matrix X + Y in which each element is the sum of the corresponding elements of X and Y.
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8. Adding Matrices - example
A toy company has plants in Boston, Chicago, and Seattle that manufacture toy phones and calculators. The following matrices show the per-item production costs for the three plants:
Use matrix addition to determine the firm’s total per-item costs.
Boston
Chicago
Seattle
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9. Adding Matrices - example
A toy company has plants in Boston, Chicago, and Seattle that manufacture toy phones and calculators. The following matrices show the per-item production costs for the three plants:
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10. Adding Matrices - example
A toy company has plants in Boston, Chicago, and Seattle that manufacture toy phones and calculators. The following matrices show the per-item production costs for the three plants:
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11. 2.3 Addition and Subtraction of Matrices
Additive Inverse
The additive inverse (or negative) of a matrix X is the matrix –X in which each element is the additive inverse of the corresponding element of X.
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12. 2.3 Addition and Subtraction of Matrices
Zero Matrix
The sum of matrices X and –X is a zero matrix
A matrix whose elements are all zeros.
If O is an m x n zero matrix, and A is any m  n matrix, then
Zero matrix – Additive identity matrix
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13. 2.3 Addition and Subtraction of Matrices
Subtracting Matrices
The difference between two m  n matrices X and Y is the m  n matrix X  Y (or Y  X) in which each element is found by subtracting the corresponding elements of X and Y.
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14. 2.3 Addition and Subtraction of Matrices
Motorcycle Helmets The following table shows the percentage of motorcyclists in various regions of the country who used helmets compliant with federal safety regulations and the percentage who used helmets that were not compliant in two recent years.
2008
Compliant
Noncompliant
Northeast 45 8
Midwest 67 16
South 61 14
West 71 5
2009
Compliant
Noncompliant
Northeast 61 15
Midwest 67 8
South 65 6
West 83 4
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15. 2.3 Addition and Subtraction of Matrices
Write two matrices for the 2008 and 2009 helmet usage.
Use the two matrices to form a matrix showing the change in helmet usage from 2008 to 2009.
2008
Compliant
Noncompliant
Northeast 45 8
Midwest 67 16
South 61 14
West 71 5
2009
Compliant
Noncompliant
Northeast 61 15
Midwest 67 8
South 65 6
West 83 4
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16. 2.3 Addition and Subtraction of Matrices
Write two matrices for the 2008 and 2009 helmet usage.
Use the two matrices to form a matrix showing the change in helmet usage from 2008 to 2009.
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17. 2.4 Multiplication of Matrices
Product of a Matrix and a Scaler (real number)
The product of a scaler k and a matrix X is the matrix kX, each of whose elements is k times the corresponding element of X.
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18. 2.4 Multiplication of Matrices
The product AB of an m  n matrix A and an n  k matrix B is found as follows.
Multiply each element of the first row of A by the corresponding element of the first column of B. The sum of these n products is the first row, first column of AB.
Multiply each element of the first row of A by the corresponding element of the second column of B. The sum of these n products is the first row, second column of AB.
Example...
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19. Multiplication of MatricesAB A B
The product AB of two matrices A and B can be found only if the number of columns of A is the same as the number of rows of B.
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20. Now You Try
Find the matrix product.
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21. Try another; pg. 105, #43 Quantity Price (in dollars) Paper Tape
Binders
Memo Pads
Pens
Department 1 10 4 3 5 6
Department 2 7 2 8
Department 3 1
Department 4
Price (in dollars)
Supplier A
Supplier B
Paper 2 3
Tape 1
Binders 4
Memo Pads
Pens
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22. Try another; pg. 105, #43
Write the information in the “Quantity” table as a 4  5 matrix Q.
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23. Try another; pg. 105, #43
Write the information in the “Price” table as a 5  2 matrix P.
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24. Try another; pg. 105, #43
Find the product QP (QP will be a 4  2 matrix)
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25. Using Microsoft Excel Array 1 is a 4 x 5 (m x n) matrix
Array 2 is a 5 x 2 (n x k) matrix
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26. Using Microsoft Excel Highlight m  k cells. Enter the function,
m = rows in array 1
k = columns in array 2
Enter the function,
=MMULT(array 1,array 2)
Select Ctrl + Shift + Enter
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27. Microsoft Excel Highlight m x k cells. (m = 4, k = 2)
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28. Microsoft Excel Highlight m x k cells. Highlight m x k cells.
(m = 4, k = 2)
Enter the function,
=MMULT(array 1,array 2)
Highlight m x k cells.
(m = 4, k = 2)
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29. Microsoft Excel Highlight m x k cells. (m = 4, k = 2)
Enter the function,
=MMULT(array 1,array 2)
Select Ctrl + Shift + Enter
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30. Microsoft Excel
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31. Microsoft Excel
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32. Microsoft Excel
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33. 2.5 Matrix Inverses Comparable to the reciprocal of a real number.
If A is an n x n matrix, A-1 is the multiplicative inverse of matrix A. (A-1 does not mean 1/A)
Just as the real number 1 is the multiplicative identity for real numbers, I is defined as the multiplicative identity matrix.
Therefore,
and,
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34. Multiplicative Identity Matrix I
2 x 2 identity matrix
3 x 3 identity matrix
4 x 4 identity matrix
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35. Multiplicative Identity Matrix I
AI = A
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36. Microsoft Excel Highlight n x n cells. Enter the function,
=MINVERSE(array)
Select Ctrl + Shift + Enter
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37. Microsoft Excel
Example:
Find A – 1
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38. Microsoft Excel
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39. Microsoft Excel
Highlight n  n cells
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40. Microsoft Excel Highlight n  n cells Enter the function,
=MINVERSE(array)
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41. Microsoft Excel Highlight n  n cells Enter the function,
=MINVERSE(array)
Select Ctrl + Shift + Enter
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42. Systems of Linear Equations
A system of linear equations is a set of n linear equations in k variables (or unknowns) that are solved together.
The simplest linear system is one with 2 equations in 2 variables.
A solution of a system is a solution that satisfies all the equations in the system.
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43. Solving a 22 System of Linear Equations
Example:
Three methods
Graph the lines and identify the intersection (if any)
Substitution
Elimination
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44. Graphing Method
(x, y)
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45. Substitution
Solve the first equation for y
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46. Substitution Solve the first equation for y
Substitute this expression for y in the second equation Solve for x
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47. Substitution Solve the first equation for y
Substitute this expression for y in the second equation Solve for x
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48. Substitution Solve the first equation for y
Substitute this expression for y in the second equation Solve for x
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49. Substitution Substitute x = 3 in either equation to solve for y.
Solution: (3, 2)
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50. Elimination
In systems of equations where the coefficients of terms containing the same variable are opposites, the elimination method can be applied by adding the equations. If the coefficients of those terms are the same, the elimination method can be applied by subtracting the equations.
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51. Elimination
Multiply the first equation by 4 and the second equation by 3, so the coefficients of y are negatives of each other.
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52. Elimination
Any solution of this system must also be the solution of the sum of the two equations
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53. Elimination
Any solution of this system must also be the solution of the sum of the two equations
Substitute x = 3 in either equation to solve for y.
Solution: (3, 2)
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54. Solving Systems of Equations With Matrix Inverses
Most practical in solving several systems that have the same variable matrix but different constants.
Write the system as a matrix equation AX = B, where
A is the matrix of the coefficients of the variables,
X is the matrix of the variables,
B is the matrix of the constants.
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55. Solving Systems of Equations With Matrix Inverses
Example: Consider a system of 3 equations in 3 variables (x, y, z)
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56. Solving Systems of Equations With Matrix Inverses
Example: Consider a system of 3 equations in 3 variables (x, y, z)
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57. Solving Systems of Equations With Matrix Inverses
Solve the matrix equation AX = B
Given: A-1A = I, and IX = X
Multiply both sides by A-1
Associative property
Multiplication inverse property
Identity property
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58. Solving Systems of Equations With Matrix Inverses
Solve the matrix equation AX = B
Given: A-1A = I, and IX = X
Multiply both sides by A-1
Associative property
Multiplication inverse property
Identity property
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59. Solving Systems of Equations With Matrix Inverses
To solve a system of equations AX = B, where A is the matrix of coefficients, X is the matrix of variables, and B is the matrix of constants, first find A-1. Then X = A-1B.
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60. Example Solve the following system using matrix notation:
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61. Example
Write each equation in proper form.
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62. Example Proper form of the system: Terms with variables on the left;
constants on the right;
variables in the same order in each equation.
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63. Example
becomes
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64. Example Write each equation in proper form.
Write the corresponding matrix equation, AX=B
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65. Example Write each equation in proper form.
Write the corresponding matrix equation, AX=B
Use Microsoft Excel to:
calculate the inverse of the coefficient matrix (A1)
multiply A1 by the constant matrix (B) to find the variable matrix (X).
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66. Excel
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67. Excel
Highlight 3  3 cells
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68. Excel
=MINVERSE(array)
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69. Excel
Ctrl + Shift + Enter
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70. Excel
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71. Excel
Highlight 3  1 cells
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72. Excel
=MMULT(array1,array2)
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73. Excel
Ctrl + Shift + Enter
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74. Excel
= X
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75. Example
x = 1, y = -3, z = 2
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76. Example
An electronics company produces transistors, resistors, and computer chips. Each transistor requires 3 units of copper, 1 unit of zinc, and 2 units of glass. Each resistor requires 3, 2, and 1 units of the three materials, and each computer chip requires 2, 1, and 2. How many of each product can be made with 810 units of copper, 410 units of zinc, and 490 units of glass?
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77. Transistor (x) Resistor (y) Comp. Chip (z) Copper 3 2 Zinc 1 Glass
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78. Transistor (x) Resistor (y) Comp. Chip (z) Copper 3 2 Zinc 1 Glass
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79. Transistor (x) Resistor (y) Comp. Chip (z) Copper 3 2 Zinc 1 Glass
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80. ECO B. Potter

81. Microsoft Excel
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82. Microsoft Excel
Use MINVERSE(b1:d3)
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83. Microsoft Excel
Use MMULT(b5:d7,g1:g3)
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84. 100 transistors, 110 resistors, and 90 computer chips can be made
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85. Now You Try
Pretzels cost $3 per pound, dried fruit $4 per pound, and nuts $8 per pound. How many pounds of each should be used to produce 140 pounds of trail mix costing $6 per pound in which there is twice as much pretzels (by weight) than dried fruit?
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86. Now You Try Let: x = the number of pounds of pretzels
y = the number of pounds of dried fruit
z = the number of pounds of nuts
The system of equations is:
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87. 2.6 Input-Output Models
Matrix models used for studying the interdependencies in an economy.
Developed by Wassily Leontief (1906 – 1909)
1973 Nobel prize in economics
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88. Input-Output Models
In practice, very complicated with many variables.
In an economy with n basic commodities, the production of each relies on inputs of the other commodities.
Example: Oil to run machinery to plant and harvest wheat.
Input-Output Matrix – shows the amounts of each commodity used in the production of one unit of each commodity.
Deal with the production and flow of goods in an economy.
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89. Input-Output Models
Input-Output matrix - simplified economy with three commodity categories (Agriculture, Manufacturing, Transportation).
Agriculture
Manufacturing
Transportation
1/3
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90. Input-Output Models
The amount of each commodity needed to produce one unit of agriculture.
Input-Output matrix - simplified economy with three commodity categories (Agriculture, Manufacturing, Transportation).
Agriculture
Manufacturing
Transportation
1/3
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91. Input-Output Models
The amount of each commodity needed to produce one unit of manufacturing.
Input-Output matrix - simplified economy with three commodity categories (Agriculture, Manufacturing, Transportation).
Agriculture
Manufacturing
Transportation
1/3
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92. Input-Output Models
The amount of each commodity needed to produce one unit of transportation.
Input-Output matrix - simplified economy with three commodity categories (Agriculture, Manufacturing, Transportation).
Agriculture
Manufacturing
Transportation
1/3
= A
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93. Input-Output Models
Input-Output matrix (A)
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94. Input-Output Models
If the economy produces:
60 units of agriculture
52 units of manufacturing
48 units of transportation
Production matrix (X) – Column matrix that gives the amount of each commodity produced by the economy:
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95. Input-Output Models Input-Output matrix (A), Production matrix (X)
¼ unit of agriculture is used to produce each unit of manufacturing.
52 units of manufacturing are produced.
¼  52 = 13 units of agriculture are used in the production of manufacturing
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96. Input-Output Models Input-Output matrix (A), Production matrix (X)
1/3 unit of agriculture is used to produce each unit of transportation.
48 units of transportation are produced.
1/3  48 = 16 units of agriculture are used in the production of transportation.
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97. Input-Output Models Input-Output matrix (A), Production matrix (X)
Therefore, = 29 units of agriculture are used for “production” in the economy
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98. Input-Output Models Input-Output matrix (A), Production matrix (X)
Since A gives the amount of each commodity needed to produce 1 unit of each,
and X gives the number of units of each commodity produced,
AX gives the amount of each commodity used in the production process.
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99. Input-Output Models Input-Output matrix (A), Production matrix (X)
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100. Input-Output Models Using Excel: Use =MMULT(B1:D3,G1:G3)
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101. Input-Output Models Input-Output matrix (A), Production matrix (X)
29 units of agriculture, 42 units of manufacturing, 28 units of transportation are used to produce 60, 52, and 48 units of each.
Remainder is used to satisfy the demand outside the production system.
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102. Input-Output Models
Therefore, production of 60 units of agriculture, 52 units of manufacturing, and 48 units of transportation would satisfy a demand of 31, 10, and 20 units of the commodities.
Demand Matrix (D) – Represents the demand for the various commodities outside the production process.
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103. Input-Output Models
If matrix I – A has an inverse, then
In practice, A and D are known and X must be found.
Where, X = amounts of production needed to satisfy the demands (D)
Identity Property
Distributive Property
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104. Input-Output Models
What should production of each commodity be to satisfy demands for 516 units of agriculture, 258 units of manufacturing, and 129 units of transportation?
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105. Input-Output Models
What should production of each commodity be to satisfy demands for 516 units of agriculture, 258 units of manufacturing, and 129 units of transportation?
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106. Input-Output Models
Excel
Use =MINVERSE(B5:D7)
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107. Input-Output Models
What should production of each commodity be to satisfy demands for 516 units of agriculture, 258 units of manufacturing, and 129 units of transportation?
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108. Input-Output Models
What should production of each commodity be to satisfy demands for 516 units of agriculture, 258 units of manufacturing, and 129 units of transportation?
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109. Input-Output Models
Excel
Use =MMULT(B5:D7,G1:G3)
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110. Input-Output Models
Production of 921 units of agriculture, 862 units of manufacturing, and 575 units if transportation is needed to satisfy demands of 516, 258, and 129.
What should production of each commodity be to satisfy demands for 516 units of agriculture, 258 units of manufacturing, and 129 units of transportation?
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111. 
Chapter 2
End
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