Systems of Linear Equations and Matrices Chapter 2 ECO 3401 - B. Potter
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 ECO 3401 - B. Potter
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). ECO 3401 - B. Potter
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. ECO 3401 - B. Potter
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. ECO 3401 - B. Potter
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 ECO 3401 - B. Potter
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. ECO 3401 - B. Potter
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 ECO 3401 - B. Potter
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: ECO 3401 - B. Potter
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: ECO 3401 - B. Potter
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. ECO 3401 - B. Potter
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 ECO 3401 - B. Potter
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. ECO 3401 - B. Potter
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 ECO 3401 - B. Potter
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 ECO 3401 - B. Potter
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. ECO 3401 - B. Potter
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. ECO 3401 - B. Potter
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... ECO 3401 - B. Potter
Multiplication of Matrices AB 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. ECO 3401 - B. Potter
Now You Try Find the matrix product. ECO 3401 - B. Potter
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 ECO 3401 - B. Potter
Try another; pg. 105, #43 Write the information in the “Quantity” table as a 4 5 matrix Q. ECO 3401 - B. Potter
Try another; pg. 105, #43 Write the information in the “Price” table as a 5 2 matrix P. ECO 3401 - B. Potter
Try another; pg. 105, #43 Find the product QP (QP will be a 4 2 matrix) ECO 3401 - B. Potter
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 ECO 3401 - B. Potter
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 ECO 3401 - B. Potter
Microsoft Excel Highlight m x k cells. (m = 4, k = 2) ECO 3401 - B. Potter
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) ECO 3401 - B. Potter
Microsoft Excel Highlight m x k cells. (m = 4, k = 2) Enter the function, =MMULT(array 1,array 2) Select Ctrl + Shift + Enter ECO 3401 - B. Potter
Microsoft Excel ECO 3401 - B. Potter
Microsoft Excel ECO 3401 - B. Potter
Microsoft Excel ECO 3401 - B. Potter
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, ECO 3401 - B. Potter
Multiplicative Identity Matrix I 2 x 2 identity matrix 3 x 3 identity matrix 4 x 4 identity matrix ECO 3401 - B. Potter
Multiplicative Identity Matrix I AI = A ECO 3401 - B. Potter
Microsoft Excel Highlight n x n cells. Enter the function, =MINVERSE(array) Select Ctrl + Shift + Enter ECO 3401 - B. Potter
Microsoft Excel Example: Find A – 1 ECO 3401 - B. Potter
Microsoft Excel ECO 3401 - B. Potter
Microsoft Excel Highlight n n cells ECO 3401 - B. Potter
Microsoft Excel Highlight n n cells Enter the function, =MINVERSE(array) ECO 3401 - B. Potter
Microsoft Excel Highlight n n cells Enter the function, =MINVERSE(array) Select Ctrl + Shift + Enter ECO 3401 - B. Potter
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. ECO 3401 - B. Potter
Solving a 22 System of Linear Equations Example: Three methods Graph the lines and identify the intersection (if any) Substitution Elimination ECO 3401 - B. Potter
Graphing Method (x, y) ECO 3401 - B. Potter
Substitution Solve the first equation for y ECO 3401 - B. Potter
Substitution Solve the first equation for y Substitute this expression for y in the second equation. Solve for x ECO 3401 - B. Potter
Substitution Solve the first equation for y Substitute this expression for y in the second equation. Solve for x ECO 3401 - B. Potter
Substitution Solve the first equation for y Substitute this expression for y in the second equation. Solve for x ECO 3401 - B. Potter
Substitution Substitute x = 3 in either equation to solve for y. Solution: (3, 2) ECO 3401 - B. Potter
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. ECO 3401 - B. Potter
Elimination Multiply the first equation by 4 and the second equation by 3, so the coefficients of y are negatives of each other. ECO 3401 - B. Potter
Elimination Any solution of this system must also be the solution of the sum of the two equations ECO 3401 - B. Potter
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) ECO 3401 - B. Potter
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. ECO 3401 - B. Potter
Solving Systems of Equations With Matrix Inverses Example: Consider a system of 3 equations in 3 variables (x, y, z) ECO 3401 - B. Potter
Solving Systems of Equations With Matrix Inverses Example: Consider a system of 3 equations in 3 variables (x, y, z) ECO 3401 - B. Potter
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 ECO 3401 - B. Potter
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 ECO 3401 - B. Potter
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. ECO 3401 - B. Potter
Example Solve the following system using matrix notation: ECO 3401 - B. Potter
Example Write each equation in proper form. ECO 3401 - B. Potter
Example Proper form of the system: Terms with variables on the left; constants on the right; variables in the same order in each equation. ECO 3401 - B. Potter
Example becomes ECO 3401 - B. Potter
Example Write each equation in proper form. Write the corresponding matrix equation, AX=B ECO 3401 - B. Potter
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 (A1) multiply A1 by the constant matrix (B) to find the variable matrix (X). ECO 3401 - B. Potter
Excel ECO 3401 - B. Potter
Excel Highlight 3 3 cells ECO 3401 - B. Potter
Excel =MINVERSE(array) ECO 3401 - B. Potter
Excel Ctrl + Shift + Enter ECO 3401 - B. Potter
Excel ECO 3401 - B. Potter
Excel Highlight 3 1 cells ECO 3401 - B. Potter
Excel =MMULT(array1,array2) ECO 3401 - B. Potter
Excel Ctrl + Shift + Enter ECO 3401 - B. Potter
Excel = X ECO 3401 - B. Potter
Example x = 1, y = -3, z = 2 ECO 3401 - B. Potter
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? ECO 3401 - B. Potter
Transistor (x) Resistor (y) Comp. Chip (z) Copper 3 2 Zinc 1 Glass ECO 3401 - B. Potter
Transistor (x) Resistor (y) Comp. Chip (z) Copper 3 2 Zinc 1 Glass ECO 3401 - B. Potter
Transistor (x) Resistor (y) Comp. Chip (z) Copper 3 2 Zinc 1 Glass ECO 3401 - B. Potter
ECO 3401 - B. Potter
Microsoft Excel ECO 3401 - B. Potter
Microsoft Excel Use MINVERSE(b1:d3) ECO 3401 - B. Potter
Microsoft Excel Use MMULT(b5:d7,g1:g3) ECO 3401 - B. Potter
100 transistors, 110 resistors, and 90 computer chips can be made ECO 3401 - B. Potter
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? ECO 3401 - B. Potter
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: ECO 3401 - B. Potter
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 ECO 3401 - B. Potter
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. ECO 3401 - B. Potter
Input-Output Models Input-Output matrix - simplified economy with three commodity categories (Agriculture, Manufacturing, Transportation). Agriculture Manufacturing Transportation ¼ 1/3 ½ ECO 3401 - B. Potter
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 ¼ 1/3 ½ ECO 3401 - B. Potter
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 ¼ 1/3 ½ ECO 3401 - B. Potter
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 ¼ 1/3 ½ = A ECO 3401 - B. Potter
Input-Output Models Input-Output matrix (A) ECO 3401 - B. Potter
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: ECO 3401 - B. Potter
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 ECO 3401 - B. Potter
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. ECO 3401 - B. Potter
Input-Output Models Input-Output matrix (A), Production matrix (X) Therefore, 13 + 16 = 29 units of agriculture are used for “production” in the economy ECO 3401 - B. Potter
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. ECO 3401 - B. Potter
Input-Output Models Input-Output matrix (A), Production matrix (X) ECO 3401 - B. Potter
Input-Output Models Using Excel: Use =MMULT(B1:D3,G1:G3) ECO 3401 - B. Potter
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. ECO 3401 - B. Potter
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. ECO 3401 - B. Potter
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 ECO 3401 - B. Potter
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? ECO 3401 - B. Potter
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? ECO 3401 - B. Potter
Input-Output Models Excel Use =MINVERSE(B5:D7) ECO 3401 - B. Potter
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? ECO 3401 - B. Potter
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? ECO 3401 - B. Potter
Input-Output Models Excel Use =MMULT(B5:D7,G1:G3) ECO 3401 - B. Potter
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? ECO 3401 - B. Potter
Chapter 2 End ECO 3401 - B. Potter