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Chapter 2 Matrices Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 1 of 90 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 Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 2 of 90 2.1 Systems of Linear Equations with Unique Solutions 1. 2. 3. 4. 5. 6. 7. 8. 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 Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 3 of 90 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: x 125 y Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel 50 z 50. Copyright © 2014 Pearson Education, Inc. 4 of 90 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 5 of 90 Elementary Row Operation 1 Elementary Row Operation 1 Interchange any two equations. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 6 of 90 Example Elementary Row Operations 1 Rearrange the equations of the system 3 x 6 x y z 0 y z 6 z 3 so that all the equations containing x are on top. 6 x [ R1 ] [ R3 ] 3x Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel z 3 y y z 6 z 0 Copyright © 2014 Pearson Education, Inc. 7 of 90 Elementary Row Operation 2 Elementary Row Operation 2 Multiply an equation by a nonzero number. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 8 of 90 Example Elementary Row Operation 2 Multiply the first row of the system 6 x 3 x z y y z 6 z 0 so that the coefficient of x is 1. x 1 [ R ] 6 1 3x Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel 3 y y 16 z z z 1 Copyright © 2014 Pearson Education, Inc. 2 6 0 9 of 90 Elementary Row Operation 3 Elementary Row Operation 3 Change an equation by adding to it a multiple of another equation. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 10 of 90 Example Elementary Row Operation 3 Add a multiple of one row to another to change x 3x y y 16 z z z 1 2 6 0 so that only the first equation has an x term. x [ R2 ] 3[ R1 ] Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel y y 16 z 32 z z 1 2 9 2 0 Copyright © 2014 Pearson Education, Inc. 11 of 90 Gaussian Elimination Method Gaussian Elimination Method transforms a system of linear equations into diagonal form by repeated applications of the three elementary row operations. 1. Rearrange the equations in any order. 2. Multiply an equation by a nonzero number. 3. Change an equation by adding to it a multiple of another equation. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 12 of 90 Example Gaussian Elimination Method Continue Gaussian Elimination to transform into diagonal form x 1 z 1 x 1[ R2 ] 6 y 32 z y z y y Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel 16 z 32 z z 2 9 2 0. 1 2 9 2 0 Copyright © 2014 Pearson Education, Inc. 13 of 90 Example Gaussian Elimination (2) x y y 16 z 32 z z x 2 [ R3 ] 1[ R2 ] 9 2 0 x 2 [ R ] 3 5 Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel y 16 z 32 z z 16 z 32 z 52 z 1 y 1 2 9 2 9 2 1 2 9 2 9 5 Copyright © 2014 Pearson Education, Inc. 14 of 90 Example Gaussian Elimination ( 3) x y 16 z 32 z z 1 2 9 2 9 5 x [ R2 ] 3 [ R3 ] 2 [ R1 ] 1 [ R3 ] 6 x y 16 z y z 1 2 9 5 9 2 4 5 9 5 z 9 5 The solution is (x,y,z) = (4/5,-9/5,9/5). Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 15 of 90 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 16 of 90 Example Matrix Form of an Equation Write the augmented matrix for the system 3x 6 y 9 z 0 4 x 6 y 8 z 4 2 x y z 7. 3 6 9 0 4 6 8 4 2 1 1 7 Note: The vertical line separates numbers that are on opposite sides of the equal sign. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 17 of 90 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: 1 0 0 * 0 1 0 * 0 0 1 * Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 18 of 90 Example Gauss–Jordan Elimination We proceed one column at a time: 3 6 9 0 4 6 8 4 2 1 1 7 1 2 3 0 0 2 4 4 2 1 1 7 1 2 3 0 0 1 2 2 0 5 7 7 1 2 3 0 4 6 8 4 2 1 1 7 1 R 3 1 R2 ( 4) R1 1 2 3 0 R3 2 R1 0 2 4 4 0 5 7 7 1 0 1 4 0 1 2 2 R1 2 R2 0 5 7 7 Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel 1 R 2 2 R3 5 R2 Copyright © 2014 Pearson Education, Inc. 19 of 90 Example Gauss–Jordan Elimination Continuing: 1 0 1 4 0 1 2 2 0 0 3 3 1 0 1 4 3 0 1 2 2 0 0 1 1 1 0 0 3 0 1 2 2 0 0 1 1 1 0 0 3 R2 2 R3 0 1 0 0 0 0 1 1 1 R3 Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel R1 1R3 Copyright © 2014 Pearson Education, Inc. 20 of 90 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, Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel z = 1. Copyright © 2014 Pearson Education, Inc. 21 of 90 Using Spreadsheet to Solve System Use a spreadsheet to solve 3 x 6 x y z y z 6 z 3. 0 Enter the augmented matrix into your spreadsheet. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 22 of 90 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 23 of 90 Spreadsheet - Entering Equations in Solver A sample set up of the constraints (equations) in Excel for Solver. The second constraint is shown. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 24 of 90 Spreadsheet - Using Solver - Setup Complete setup for Solver. Solution is calculated once Solve is clicked. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 25 of 90 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 26 of 90 Summary Section 2.1 - 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). Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 27 of 90 Summary Section 2.1 - 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 28 of 90 2.2 General Systems of Linear Equations 1. 2. 3. 4. 5. Pivot a Matrix Gaussian Elimination Method Infinitely Many Solutions Inconsistent System Geometric Representation of System Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 29 of 90 Pivot a Matrix Method To pivot a matrix about a given nonzero entry: 1. Transform the given entry into a one; 2. Transform all other entries in the same column into zeros. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 30 of 90 Example Pivot a Matrix Pivot the matrix about the circled element. 18 6 15 5 2 4 18 6 15 5 2 4 3 1 R 6 1 5 5 2 2 4 1 5 3 1 R2 2 R1 2 1 0 1 Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 31 of 90 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 32 of 90 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 33 of 90 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 34 of 90 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 35 of 90 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 36 of 90 Example Infinitely Many Solutions Find all solutions of 2 2 4 8 1 1 2 2 1 5 2 2 2x 2 y 4z 8 x y 2z 2 x 5 y 2 z 2. 12 R 1 1 2 4 0 2 0 2 R2 ( 1) R1 R3 1 R1 0 6 0 6 12 R 1 0 2 3 0 1 0 1 R1 ( 1) R2 R3 6 R2 0 0 0 0 2 Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel 1 General Solution z = any real number x = 3 - 2z y=1 Copyright © 2014 Pearson Education, Inc. 37 of 90 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 38 of 90 Example Inconsistent System Find all solutions of 1 1 1 3 1 1 1 5 2 4 4 1 y z 3 x y z 5 x 2 x 4 y 4 z 1. 1 1 1 3 0 2 2 2 R2 ( 1) R1 R3 2 R1 0 2 2 7 12 R 1 0 0 4 0 1 1 1 R1 (1) R2 R3 2 R2 0 0 0 5 2 Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Because of the last row, the system is inconsistent. Copyright © 2014 Pearson Education, Inc. 39 of 90 Summary Section 2.2 - 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 40 of 90 Summary Section 2.2 - 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 41 of 90 Summary Section 2.2 - Part 3 If one row of the reduced augmented matrix has the form 0 0 0 … 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 42 of 90 2.3 Arithmetic Operations on Matrices 1. 2. 3. 4. 5. 6. 7. 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 Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 43 of 90 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 44 of 90 Example Definition of Matrix 4 1 5 is a 2x3 matrix. 3 0 7 The entry a1,2 = -1. The entry a2,3 = 7. 3 9 3 9 7 0 7 0 2 2 2 2 Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel 3 9 3 7 2 9 0 2 7 0 2 2 Copyright © 2014 Pearson Education, Inc. 45 of 90 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 46 of 90 Example Column, Row & Square Matrix 4 1 3 7 is a 2x2 matrix and a square matrix. 2 8 7 1 is a 1x4 matrix and a row matrix. 2 1 2 is a 3x1 matrix and a column matrix. 3.4 Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 47 of 90 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 48 of 90 Example Addition & Subtraction 2 1 3 1 4 7 1 3 10 4 0 5 8 3 2 12 3 3 2 1 3 1 4 7 3 5 4 4 0 5 8 3 2 4 3 7 1 8 2 1 3 4 0 5 4 3 is not defined. 7 2 Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 49 of 90 Multiplying Row Matrix to Column Matrix If A is a row matrix and B is a column matrix, then we can form the product AB provided that the two matrices have the same length. The product AB is a 1x1 matrix obtained by multiplying corresponding entries of A and B and then forming the sum. a1 a2 b1 b an 2 a1b1 a2b2 bn Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel anbn Copyright © 2014 Pearson Education, Inc. 50 of 90 Example Multiplying Row to Column 3 2 2 1 3 5 2 3 1 2 3 5 7 3 4 0 2 1 2 5 Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel is not defined. Copyright © 2014 Pearson Education, Inc. 51 of 90 Matrix Multiplication If A is an mxn matrix and B is an nxq matrix, then we can form the product AB. The product AB 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 AB is formed by multiplying the ith row of A and jth column of B. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 52 of 90 Example Matrix Multiplication 3 2 0 2 1 3 3 0 2 2 1 2 5 3 1 7 12 -5 2 -19 0 3 2 0 2 1 2 2 1 3 is not defined. 3 0 2 5 3 1 Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 53 of 90 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: 1 0 I2 0 1 Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel 1 0 0 I 3 0 1 0 . 0 0 1 Copyright © 2014 Pearson Education, Inc. 54 of 90 Example Identity Matrix 1 0 2 1 3 2 1 3 0 1 3 0 2 3 0 2 1 0 0 2 1 3 2 1 3 0 1 0 3 0 2 3 0 2 0 0 1 For all nxn matrices A, In A = A In = A. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 55 of 90 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 56 of 90 Example Matrix Equation Write the following system as a matrix equation 4x 7 y 9 3x 2 y 5. x y constants 4 7 x 9 Equation 2 3 2 y 5 Equation 1 Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 57 of 90 Summary Section 2.3 - 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 58 of 90 Summary Section 2.3 - 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.) Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 59 of 90 2.4 The Inverse of a Matrix 1. 2. 3. 4. Inverse of A Inverse of a 2x2 Matrix Matrix With No Inverse Solving a Matrix Equation Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 60 of 90 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 61 of 90 Example Inverse of A 1 4 Verify that 11 11 is the inverse of 3 2 11 11 2 1 3 4 . 1 4 2 1 1 0 11 11 3 2 3 4 0 1 11 11 checks 4 1 2 1 11 11 1 0 3 4 3 0 1 2 11 11 checks Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 62 of 90 Inverse of a 2x2 To determine the inverse of a b if D = ad - bc ≠ 0, c d 1. Interchange a and d to get d b . c a 2. Change the signs of b and c to get d b c a . 3. b d D . Divide all entries by D to get D c a D D Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 63 of 90 Example Inverse of a 2x2 2 4 Find the inverse of . 3 7 D = (-2)(7) - (4)(-3) = -2 ≠ 0 7 4 7 4 2. 1. Interchange: 3 2 Change signs: 3 2 4 7 7 2 3. 2 2 2 A1 Divide: 3 2 3 1 2 2 2 Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 64 of 90 Matrix With No Inverse A matrix a b has no inverse if c d D = ad - bc = 0. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 65 of 90 Example Inverse or No Inverse Use D to determine which matrix has an inverse. 2 4 3 6 has no inverse. 2 4 3 6 has an inverse. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 66 of 90 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 67 of 90 Example Solving a Matrix Equation 2 x 4 y 2 Use a matrix equation to solve 3x 7 y 7. The matrix form of the equation is 2 4 x 2 3 7 y 7 . 1 7 2 2 7 x 2 4 2 2 y 3 7 7 3 1 7 4 2 Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 68 of 90 Summary Section 2.4 - 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 a b c d has an inverse if = ad - bc ≠ 0. If so, the inverse matrix is b d . c a Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 69 of 90 Summary Section 2.4 - 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 70 of 90 2.5 The Gauss-Jordan Method for Calculating Inverses 1. Gauss-Jordan Method for Inverses Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 71 of 90 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 72 of 90 Example Inverses 4 2 3 Find the inverse of A 8 3 5 . 7 2 4 4 2 3 1 0 0 Step 1: Step 2: 8 3 5 0 1 7 2 4 0 0 1 1 3 1 2 4 4 0 1 1 2 5 7 0 3 2 4 4 Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel 0 1 0 0 1 0 0 1 Copyright © 2014 Pearson Education, Inc. 73 of 90 Example Inverses (2) 1 0 0 1 0 0 3 0 4 4 2 1 2 1 0 1 5 3 1 4 4 2 1 1 1 0 0 2 2 1 0 1 0 3 5 4 0 0 1 5 6 4 Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Step 3: 2 2 1 1 A 3 5 4 5 6 4 Copyright © 2014 Pearson Education, Inc. 74 of 90 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.) Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 75 of 90 2.6 The Input-Output Analysis 1. 2. 3. 4. Input-Output Analysis Input-Output Matrix Final Demand Production Level Problem Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 76 of 90 Input-Output Analysis Input-output analysis is used to analyze an economy in order to meet given consumption and export demands. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 77 of 90 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 78 of 90 Input-Output Matrix - Form A typical input-output matrix looks like: Input requirements of: Industry 1 Industry 2 Industry 3 Industry 1 From Industry 2 Industry 3 . Each column gives the dollar values of the various inputs needed by an industry in order to produce $1 worth of output. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 79 of 90 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 inputoutput matrix for this economy. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 80 of 90 Example Input-Output Matrix Answer Coal Steel Electricity 0 Steel .02 Electricity .01 Coal Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel .15 .43 .03 .20 .08 .05 Copyright © 2014 Pearson Education, Inc. A 81 of 90 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: amount from industry 1 final demand amount from industry 2 . Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 82 of 90 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 83 of 90 Example Final Demand Answer For simplicity, set up the final demand matrix in billions of dollars. 2 Steel 1 D Electricity 3 Coal Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 84 of 90 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 Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 85 of 90 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 86 of 90 Example Production Level Solution (1) 1 0 0 0 .15 .43 1 -.15 -.43 I A 0 1 0 .02 .03 .20 -.02 .97 -.20 0 0 1 .01 .08 .05 -.01 -.08 .95 1 1 -.15 -.43 1.01 .20 .50 1 ( I A) -.02 .97 -.20 .02 1.05 .23 -.01 -.08 .95 .01 .09 1.08 Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 87 of 90 Example Production Level Solution (2) 1.01 .20 .50 2 3.72 1 X ( I A) D .02 1.05 .23 1 1.78 .01 .09 1.08 3 3.35 The three industries should produce $3.72 billion of coal, $1.78 billion of steel and $3.35 billion of electricity. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 88 of 90 Summary Section 2.5 - 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 89 of 90 Summary Section 2.5 - 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. Finite Mathematics & Its Applications, 11/e by Goldstein/Schneider/Siegel Copyright © 2014 Pearson Education, Inc. 90 of 90