1. Chapter 2
Matrices
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2. Outline 2.1 Systems of Linear Equations with Unique Solutions
2.2 General Systems of Linear Equations
2.3 Arithmetic Operations on Matrices
2.4 The Inverse of a Matrix
2.5 The Gauss-Jordan Method for Calculating Inverses
2.6 Input-Output Analysis
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3. 2.1 Systems of Linear Equations with Unique Solutions
Diagonal Form of a System of Equations
Elementary Row Operations
Elementary Row Operation 1
Elementary Row Operation 2
Elementary Row Operation 3
Gaussian Elimination Method
Matrix Form of an Equation
Using Spreadsheet to Solve System
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4. Diagonal Form of a System of Equations
A system of equations is in diagonal form if each variable only appears in one equation and only one variable appears in an equation.
For example:
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5. Elementary Row Operations
Elementary row operations are operations on the equations (rows) of a system that alters the system but does not change the solutions.
Elementary row operations are often used to transform a system of equations into a diagonal system whose solution is simple to determine.
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6. Elementary Row Operation 1
Elementary Row Operation 1 Interchange any two equations.
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7. Example Elementary Row Operations 1
Rearrange the equations of the system
so that all the equations containing x are on top.
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8. Elementary Row Operation 2
Elementary Row Operation 2 Multiply an equation by a nonzero number.
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9. Example Elementary Row Operation 2
Multiply the first row of the system
so that the coefficient of x is 1.
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10. Elementary Row Operation 3
Elementary Row Operation 3 Change an equation by adding to it a multiple of another equation.
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11. Example Elementary Row Operation 3
Add a multiple of one row to another to change
so that only the first equation has an x term.
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12. Gaussian Elimination Method
Gaussian Elimination Method transforms a system of linear equations into diagonal form by repeated applications of the three elementary row operations.
Rearrange the equations in any order.
Multiply an equation by a nonzero number.
Change an equation by adding to it a multiple of another equation.
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13. Example Gaussian Elimination Method
Continue Gaussian Elimination to transform into diagonal form
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14. Example Gaussian Elimination (2)
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15. Example Gaussian Elimination ( 3)
The solution is (x,y,z) = (4/5,-9/5,9/5).
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16. Matrix Form of an Equation
It is often easier to do row operations if the coefficients and constants are set up in a table (matrix, plural matrices).
Each row represents an equation.
Each column represents a variable’s coefficients except the last which represents the constants.
Such a table is called the augmented matrix of the system of equations.
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17. Example Matrix Form of an Equation
Write the augmented matrix for the system
Note: The vertical line separates numbers that are on opposite sides of the equal sign.
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18. Example Gauss–Jordan Elimination
We must use elementary row operations to transform this array into diagonal form— that is, with ones down the diagonal, and zeros everywhere else on the left:
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19. Example Gauss–Jordan Elimination
We proceed one column at a time:
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20. Example Gauss–Jordan Elimination
Continuing:
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21. Example Gauss–Jordan Elimination
The last array is in diagonal form, so we just put back the variables and read off the solution:
x = −3, y = 0, z = 1.
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22. Using Spreadsheet to Solve System
Use a spreadsheet to solve
Enter the augmented matrix into your spreadsheet.
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23. Spreadsheet - Entering Left Side of Equations
A sample set up of the left side of the equations in cells B1, B2 and B3 in Excel. The third equation is shown. The three variables’ cells are A1, A2 and A3.
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24. Spreadsheet - Entering Equations in Solver
A sample set up of the constraints (equations) in Excel for Solver. The second constraint is shown.
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25. Spreadsheet - Using Solver - Setup
Complete setup for Solver.
Solution is calculated once Solve is clicked.
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26. Spreadsheet - Using Solver
A sample solution (in column A) in Excel using Solver.
The solution is
x = 0.8
y = -1.8
z = 1.8.
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27. Summary Section Part 1
The three elementary row operations for a system of linear equations (or a matrix) are as follows:
Rearrange the equations (rows) in any order;
Multiply an equation (row) by a nonzero number;
Change an equation (row) by adding to it a multiple of another equation (row).
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28. Summary Section Part 2
When an elementary row operation is applied to a system of linear equations (or an augmented matrix) the solutions remain the same. The Gaussian Elimination Method is a systematic process that applies a sequence of elementary row operations until the solutions can be easily obtained.
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29. 2.2 General Systems of Linear Equations
Pivot a Matrix
Gaussian Elimination Method
Infinitely Many Solutions
Inconsistent System
Geometric Representation of System
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30. Pivot a Matrix Method To pivot a matrix about a given nonzero entry:
Transform the given entry into a one;
Transform all other entries in the same column into zeros.
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31. Example Pivot a Matrix Pivot the matrix about the circled element.
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32. Gaussian Elimination Method
Gaussian Elimination Method to Transform a System of Linear Equations into Diagonal Form
1. Write down the matrix corresponding to the linear system.
2. Make sure that the first entry in the first column is nonzero. Do this by interchanging the first row with one of the rows below it, if necessary.
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33. Gaussian Elimination Method (2)
Gaussian Elimination Method to Transform a System of Linear Equations into Diagonal Form
3. Pivot the matrix about the first entry in the first column.
4. Make sure that the second entry in the second column is nonzero. Do this by interchanging the second row with one of the rows below it, if necessary.
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34. Gaussian Elimination Method (3)
Gaussian Elimination Method to Transform a System of Linear Equations into Diagonal Form
5. Pivot the matrix about the second entry in the second column.
6. Continue in this manner until the left side of the matrix is in diagonal form.
7. Write the system of linear equations corresponding to the matrix.
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35. Infinitely Many Solutions
When a linear system cannot be completely diagonalized,
1. Apply the Gaussian elimination method to as many columns as possible. Proceed from left to right, but do not disturb columns that have already been put into proper form. As much as possible, each row should have a 1 in its leftmost nonzero entry. (Such a 1 is called a leading 1.) The column for each leading 1 should be to the right of the columns for the leading 1s in the rows above it.
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36. Infinitely Many Solutions (2)
2. Variables corresponding to columns not in proper form can assume any value.
3. The other variables can be expressed in terms of the variables of step 2.
4. This will give the general form of the solution.
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37. Example Infinitely Many Solutions
Find all solutions of
General Solution
z = any real number
x = 3 - 2z
y = 1
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38. Inconsistent System
When using the Gaussian Elimination Method, if a row of zeros occurs to the left of the vertical line and a nonzero number is to the right of the vertical line in the same row, then the system has no solution and is said to be inconsistent.
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39. Example Inconsistent System
Find all solutions of
Because of the last row, the system is inconsistent.
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40. Summary Section Part 1
The process of pivoting on a specific entry of a matrix is to apply a sequence of elementary row operations so that the specific entry becomes 1 and the other entries in its column become 0.
To apply the Gaussian Elimination Method, proceed from left to right and perform pivots on as many columns to the left of the vertical line as possible, with the specific entries for the pivots coming from different rows.
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41. Summary Section Part 2
After an augmented matrix has been completely reduced with the Gaussian Elimination Method, all the solutions to the corresponding system of linear equations can be obtained.
If the reduced augmented matrix has a 1 in every column to the left of the vertical line, then there is a unique solution.
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42. Summary Section Part 3
If one row of the reduced augmented matrix has the form … 0 | a where a ≠ 0, then there is no solution.
Otherwise, there are infinitely many solutions. In this case, variables corresponding to columns that have not been pivoted can assume any values, and the values of the other variables can be expressed in terms of those variables.
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43. 2.3 Arithmetic Operations on Matrices
Definition of Matrix
Column, Row and Square Matrix
Addition and Subtraction of Matrices
Multiplying Row Matrix to Column Matrix
Matrix Multiplication
Identity Matrix
Matrix Equation
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44. Definition of Matrix
A matrix is any rectangular array of numbers and may be of any size.
The size of a matrix is nxk where n is the number of rows and k is the number of columns.
The entry aij refers to the number in the ith row and jth column of the matrix.
Two matrices are equal provided that they have the same size and that all their corresponding entries are equal.
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45. Example Definition of Matrix
is a 2x3 matrix.
The entry a1,2 = -1.
The entry a2,3 = 7.
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46. Column, Row and Square Matrix
A row matrix or row vector only has one row.
A column matrix or column vector only has one column.
A square matrix has the same number of rows as columns.
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47. Example Column, Row & Square Matrix
is a 2x2 matrix and a square matrix.
is a 1x4 matrix and a row matrix.
is a 3x1 matrix and a column matrix.
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48. Addition and Subtraction of Matrices
The sum A + B of two matrices A and B is defined only if A and B are two matrices of the same size. In this case A + B is the matrix formed by adding the corresponding entries of A and B.
Two matrices of the same size are subtracted by subtracting corresponding entries.
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49. Example Addition & Subtraction
is not defined.
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50. Multiplying Row Matrix to Column Matrix
If A is a row matrix and B is a column matrix, then we can form the product AB provided that the two matrices have the same length. The product AB is a 1x1 matrix obtained by multiplying corresponding entries of A and B and then forming the sum.
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51. Example Multiplying Row to Column
is not defined.
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52. Matrix Multiplication
If A is an mxn matrix and B is an nxq matrix, then we can form the product AB. The product AB is an mxq matrix whose entries are obtained by multiplying the rows of A by the columns of B. The entry in the ith row and jth column of the product AB is formed by multiplying the ith row of A and jth column of B.
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53. Example Matrix Multiplication7 12 -5
-19 2
is not defined.
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54. Identity Matrix
The identity matrix In of size n is the nxn square matrix with all zeros except for ones down the upper-left-to-lower-right diagonal.
Here are the identity matrix of sizes 2 and 3:
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55. Example Identity Matrix
For all nxn matrices A, In A = A In = A.
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56. Matrix Equation The matrix form of a system of linear equations is
AX = B
where A is the coefficient matrix whose rows correspond to the coefficients of the variables in the equations. X is the column matrix corresponding to the variables in the system. B is the column matrix corresponding to the constants on the right-hand side of the equations.
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57. Example Matrix Equation
Write the following system as a matrix equation
x y constants
Equation 1
Equation 2
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58. Summary Section Part 1
A matrix of size mxn has m rows and n columns.
Matrices of the same size can be added (or subtracted) by adding (or subtracting) corresponding elements.
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59. Summary Section Part 2
The product of an mxn and an nxr matrix is the mxr matrix whose ijth element is obtained by multiplying the ith row of the first matrix by the jth column of the second matrix. (The product of each row and column is calculated as the sum of the products of successive entries.)
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60. 2.4 The Inverse of a Matrix Inverse of A Inverse of a 2x2 Matrix
Matrix With No Inverse
Solving a Matrix Equation
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61. Inverse of A
The inverse of a square matrix A, denoted by A-1, is a square matrix with the property
A-1A = AA-1 = I,
where I is an identity matrix of the same size.
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62. Example Inverse of A Verify that is the inverse of checks checks
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63. Inverse of a 2x2 To determine the inverse of if D = ad - bc ≠ 0,
Interchange a and d to get
Change the signs of b and c to get
Divide all entries by D to get
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64. Example Inverse of a 2x2 Find the inverse of2.
Change signs:
1. Interchange: 3.
Divide:
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65. Matrix With No Inverse A matrix has no inverse if D = ad - bc = 0.
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66. Example Inverse or No Inverse
Use D to determine which matrix has an inverse.
has no inverse.
has an inverse.
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67. Solving a Matrix Equation
Solving a Matrix Equation If the matrix A has an inverse, then the solution of the matrix equation
AX = B is given by X = A-1B.
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68. Example Solving a Matrix Equation
Use a matrix equation to solve
The matrix form of the equation is
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69. = ad - bc ≠ 0. If so, the inverse matrix is
Summary Section Part 1
The inverse of a square matrix A is a square matrix A-1 with property that A-1A = I and AA-1 = I, where I is the identity matrix.
A 2x2 matrix has an inverse if
= ad - bc ≠ 0. If so, the inverse matrix is
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70. Summary Section Part 2
A system of linear equations can be written in the form AX = B, where A is a rectangular matrix of coefficients of the variables, X is a column of variables, and B is a column matrix of the constants from the right side of the system. If the matrix A has an inverse, then the solution of the equation is given by X = A-1B.
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71. 2.5 The Gauss-Jordan Method for Calculating Inverses
Gauss-Jordan Method for Inverses
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72. Gauss-Jordan Method for Inverses
Step 1: Write down the matrix A, and on its right write an identity matrix of the same size.
Step 2: Perform elementary row operations on the left-hand matrix so as to transform it into an identity matrix. These same operations are performed on the right-hand matrix.
Step 3: When the matrix on the left becomes an identity matrix, the matrix on the right is the desired inverse.
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73. Example Inverses Find the inverse of Step 1: Step 2:
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74. Example Inverses (2) Step 3:
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75. Summary Section 2.5
To calculate the inverse of a matrix by the Gauss-Jordan method, append an identity matrix to the right of the original matrix and perform pivots to reduce the original matrix to an identity matrix. The matrix on the right will then be the inverse of the original matrix. (If the original matrix cannot be reduced to an identity matrix, then the original matrix does not have an inverse.)
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76. 2.6 The Input-Output Analysis
Input-Output Matrix
Final Demand
Production Level Problem
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77. Input-Output Analysis
Input-output analysis is used to analyze an economy in order to meet given consumption and export demands.
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78. Input-Output Matrix
The economy is divided into a number of industries. Each industry produces a certain output using the outputs of other industries as inputs. This interdependence among the industries can be summarized in a matrix - an input-output matrix. There is one column for each industry’s input requirements. The entries in the column reflect the amount of input required from each of the industries.
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79. Input-Output Matrix - Form
A typical input-output matrix looks like:
Input requirements of:
Each column gives the dollar values of the various inputs needed by an industry in order to produce $1 worth of output.
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80. Example Input-Output Matrix
An economy is composed of three industries - coal, steel, and electricity. To make $1 of coal, it takes no coal, but $.02 of steel and $.01 of electricity; to make $1 of steel, it takes $.15 of coal, $.03 of steel, and $.08 of electricity; and to make $1 of electricity, it takes $.43 of coal, $.20 of steel, and $.05 of electricity. Set up the input-output matrix for this economy.
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81. Example Input-Output Matrix Answer
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82. Final Demand
The final demand on the economy is a column matrix with one entry for each industry indicating the amount of consumable output demanded from the industry not used by the other industries:
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83. Example Final Demand
An economy is composed of three industries - coal, steel, and electricity as in the previous example. Consumption (amount not used for production) is projected to be $2 billion for coal, $1 billion for steel and $3 billion for electricity. Set up the final demand matrix.
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84. Example Final Demand Answer
For simplicity, set up the final demand matrix in billions of dollars.
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85. Production Level Problem
Problem: Find the amount of production of each industry to meet the final demand of the economy. Our problem is to determine the output of each industry, X, that yields the desired amounts left over from the production process.
Answer:
X = (I - A)-1D
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86. Example Production Level Problem
An economy is composed of three industries - coal, steel, and electricity as in the previous examples. Find the output of each industry that will meet the final demand.
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87. Example Production Level Solution (1)
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88. Example Production Level Solution (2)
The three industries should produce $3.72 billion of coal, $1.78 billion of steel and $3.35 billion of electricity.
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89. Summary Section Part 1
An input-output matrix has rows and columns labeled with the different industries in an economy. The ijth entry of the matrix gives the cost of the input from the industry in row i used in the production of $1 worth of the output of industry in column j.
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90. Summary Section Part 2
If A is an input-output matrix and D is a demand matrix giving the dollar values of the outputs from various industries to be supplied to outside customers, then the matrix X = (I - A)-1D gives the amounts that must be produced by the various industries in order to meet the demand.
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