goldstein/schnieder/lay: finite math & its applications, 9e 1 of 86 chapter 2 matrices

Goldstein/Schnieder/Lay: Finite Math & Its Applications, 9e 1 of 86

Chapter 2

Matrices


Outline

2.1 Solving Systems of Linear Equations I

2.2 Solving Systems of Linear Equations II

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


2.1 Solving Systems of Linear Equations I

1. Diagonal Form of a System of Equations2. Elementary Row Operations3. Elementary Row Operation 14. Elementary Row Operation 25. Elementary Row Operation 36. Gaussian Elimination Method7. Matrix Form of an Equation8. Using Spreadsheet to Solve System


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: 50

125

50

x

y

z


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.


Elementary Row Operation 1

Elementary Row Operation 1 Rearrange the equations in any order.


Example Elementary Row Operations 1

Rearrange the equations of the system

so that all the equations containing x are on top.

0

3 6

6 3

y z

x y z

x z

[1] [3]

6 3

3 6

0

x z

x y z

y z



Elementary Row Operation 2 Multiply an equation by a nonzero number.


Example Elementary Row Operation 2

Multiply the first row of the system

so that the coefficient of x is 1.

1 [1]6

1 16 2

3 6

0

x z

x y z

y z

6 3

3 6

0

x z

x y z

y z



Elementary Row Operation 3 Change an equation by adding to it a multiple of another equation.


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.

[2] 3 [1]

1 16 2

3 92 2

0

x z

y z

y z

1 16 2

3 6

0

x z

x y z

y z


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.


Example Gaussian Elimination Method

Continue Gaussian Elimination to transform into diagonal form

1 [2]

1 16 2

3 92 2

0

x z

y z

y z

1 16 2

3 92 2

0.

x z

y z

y z


Example Gaussian Elimination (2)

2 [3]5

1 16 2

3 92 2

95

x z

y z

z

[3] 1 [2]

1 1 1 16 2 6 2

3 9 3 92 2 2 2

0 5 92 2

x z x z

y z y z

y z z


Example Gaussian Elimination ( 3)

1[1] [3]6

45

95

95

x

y

z

3[2] [3]2

1 1 1 16 2 6 2

3 9 92 2 5

9925

x z x z

y z y

zz

The solution is (x,y,z) = (4/5,-9/5,9/5).


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).

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.


Example Matrix Form of an Equation

Write the augmented matrix for the system0

3 6

6 3.

y z

x y z

x z

0 1 1 0

3 1 1 6

6 0 1 3

Note: The vertical line separates numbers that are on opposite sides of the equal sign.


Using Spreadsheet to Solve System

Use a spreadsheet to solve

Enter the augmented matrix into your spreadsheet.

0

3 6

6 3.

y z

x y z

x z


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.


Spreadsheet - Entering Equations in Solver

A sample set up of the constraints (equations) in Excel for Solver. The second constraint is shown.


Spreadsheet - Using Solver - Setup

Complete setup for Solver.

Solution is calculated once Solve is clicked.


Spreadsheet - Using SolverA sample solution (in column A) in Excel using Solver.

The solution is x = 0.8y = -1.8z = 1.8.


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).



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.


2.2 Solving Systems of Linear Equations, II

1. Pivot a Matrix

2. Gaussian Elimination Method

3. Infinitely Many Solutions

4. Inconsistent System

5. Geometric Representation of System


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.


Example Pivot a Matrix

Pivot the matrix about the circled element.

18 6 15

5 2 4

1 [1]653 118 6 15 2

5 2 4 5 2 4

[2] 2[1]53 1 2

1 0 1


Gaussian Elimination Method

Gaussian Elimination Method to Transform a System of Linear Equations into Diagonal Form1. 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.


Gaussian Elimination Method (2)

Gaussian Elimination Method to Transform a System of Linear Equations into Diagonal Form3. 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.


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.


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.

2. Variables corresponding to columns not in proper form can assume any value.


Infinitely Many Solutions (2)

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.


Example Infinitely Many Solutions

Find all solutions of 2 2 4 8

2 2

5 2 2.

x y z

x y z

x y z

General Solution

z = any real number

x = 3 - 2z

y = 1

1 [1]2[2] ( 1)[1][3] 1 [1]

2 2 4 8 1 1 2 4

1 1 2 2 0 2 0 2

1 5 2 2 0 6 0 6

1 [2]2[1] ( 1)[2][3] 6 [2]

1 0 2 3

0 1 0 1

0 0 0 0


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.


Example Inconsistent System

Find all solutions of 3

5

2 4 4 1.

x y z

x y z

x y z

Because of the last row, the system is inconsistent.

[2] ( 1)[1][3] 2 [1]

1 1 1 3 1 1 1 3

1 1 1 5 0 2 2 2

2 4 4 1 0 2 2 7

1 [2]2[1] (1)[2]

[3] 2 [2]

1 0 0 4

0 1 1 1

0 0 0 5



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.



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.



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.


2.3 Arithmetic Operations on Matrices

1. Definition of Matrix2. Column, Row and Square Matrix3. Addition and Subtraction of Matrices4. Multiplying Row Matrix to Column Matrix5. Matrix Multiplication6. Identity Matrix7. Matrix Equation


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.


Example Definition of Matrix

is a 2x3 matrix.

The entry a1,2 = -1.

The entry a2,3 = 7.

4 1 5

3 0 7

3 9 3 9

7 0 7 0

2 2 2 2

3 93 7 2

7 09 0 2

2 2


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.


Example Column, Row & Square Matrix

is a 2x2 matrix and a square matrix.4 1

3 7

2 8 7 1 is a 1x4 matrix and a row matrix.

2

12

3.4

is a 3x1 matrix and a column matrix.


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.


Example Addition & Subtraction

2 1 3 1 4 7

4 0 5 8 3 2

1 3 10

12 3 3

1 82 1 3

4 34 0 5

7 2

2 1 3 1 4 7

4 0 5 8 3 2

3 5 4

4 3 7

is not defined.


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.

1

21 2 1 1 2 2n n n

n

b

ba a a a b a b a b

b


Example Multiplying Row to Column

3

2 1 3 2

5

2 3 1 2 3 5 7

3

4 0 2 1 2

5

is not defined.


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.


Example Matrix Multiplication

7 12 -5

-19 0 2

3 2 02 1 3

2 1 23 0 2

5 3 1

is not defined.

3 2 02 1 3

2 1 23 0 2

5 3 1


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:

2 3

1 0 01 0

0 1 0 .0 1

0 0 1

I I


Example Identity Matrix

1 0 2 1 3

0 1 3 0 2

2 1 3

3 0 2

1 0 02 1 3

0 1 03 0 2

0 0 1

For all nxn matrices A, In A = A In = A.

2 1 3

3 0 2


Matrix Equation

The matrix form of a system of linear equations is

AX = Bwhere 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.


Example Matrix Equation

4 7 9

3 2 5.

x y

x y

Write the following system as a matrix equation

4 7 9

3 2 5

x

y

Equation 1

Equation 2

x y constants



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.



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.)


2.4 The Inverse of a Matrix

1. Inverse of A

2. Inverse of a 2x2 Matrix

3. Matrix With No Inverse

4. Solving a Matrix Equation


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.


Example Inverse of A

Verify that is the inverse of 4 1

11 113 211 11

2 1.

3 4

4 1 2 1 1 011 113 3 4 0 1211 11

4 12 1 1 011 1133 4 0 1211 11

checks

checks


Inverse of a 2x2

To determine the inverse of if = ad - bc ≠ 0.

1. Interchange a and d to get

2. Change the signs of b and c to get

3. Divide all entries by to get

a b

c d

.d b

c a

.d b

c a

.d b

c a


Example Inverse of a 2x2

Find the inverse of 2 4

.3 7

1

7 74 22 2 23 32 12 2 2

A

7 4

3 2

1. Interchange:

7 4

3 2

2.Change signs:

3.Divide:

= (-2)(7) - (4)(-3) = -2 ≠ 0


Matrix With No Inverse

A matrix has no inverse if a b

c d

= ad - bc = 0.


Example Inverse or No Inverse

Use to determine which matrix has an inverse.

2 4

3 6

2 4

3 6

has no inverse.

has an inverse.


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.


Example Solving a Matrix Equation

Use a matrix equation to solve 2 4 2

3 7 7.

x y

x y

The matrix form of the equation is 2 4 2

.3 7 7

x

y

1 7 22 4 2 2 723 7 7 3 7 412

x

y


A 2x2 matrix has an inverse if

= ad - bc ≠ 0. If so, the inverse matrix is


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 b

c d

.d b

c a



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.


1. Gauss-Jordan Method for Inverses

2.5 The Gauss-Jordan Method for Calculating Inverses


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.


Example Inverses

Find the inverse of

4 2 3

8 3 5 .

7 2 4

A

Step 1:

31 11 0 02 4 40 1 1 2 1 0

3 5 70 0 12 4 4

Step 2:

4 2 3 1 0 0

8 3 5 0 1 0

7 2 4 0 0 1


31 11 0 04 4 20 1 1 2 1 0

1 5 30 0 14 4 2

Example Inverses (2)

1 0 0 2 2 1

0 1 0 3 5 4

0 0 1 5 6 4

1

2 2 1

3 5 4

5 6 4

A

Step 3:


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.)


2.6 The Input-Output Analysis

1. Input-Output Analysis

2. Input-Output Matrix

3. Final Demand

4. Production Level Problem


Input-Output Analysis

Input-output analysis is used to analyze an economy in order to meet given consumption and export demands.


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.


Input-Output Matrix - Form

A typical input-output matrix looks like:

Industry 1 Industry 2 Industry 3

Industry 1

From .Industry 2

Industry 3

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.


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.


Example Input-Output Matrix Answer

Coal Steel Electricity

Coal 0 .15 .43

Steel .02 .03 .20

Electricity .01 .08 .05

A


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 .


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.


Example Final Demand Answer

For simplicity, set up the final demand matrix in billions of dollars.

Coal 2

Steel 1

Electricity 3

D


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


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.


Example Production Level Solution (1)

1 0 0 0 .15 .43

0 1 0 .02 .03 .20

0 0 1 .01 .08 .05

I A

1 -.15 -.43

-.02 .97 -.20

-.01 -.08 .95

1

1

1 -.15 -.43 1.01 .20 .50

( ) -.02 .97 -.20 .02 1.05 .23

-.01 -.08 .95 .01 .09 1.08

I A


Example Production Level Solution (2)

1

1.01 .20 .50 2 3.72

( ) .02 1.05 .23 1 1.78

.01 .09 1.08 3 3.35

X I A D

The three industries should produce $3.72 billion of coal, $1.78 billion of steel and $3.35 billion of electricity.



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.



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.

goldstein/schnieder/lay: finite math & its applications, 9e 1 of 86 chapter 2 matrices

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