`CHAPTER 1Matrix Algebra

1.1 Introduction1.2 Type of Matrices1.3 Matrix Operations1.4 Determinants 1.5 Inverse Matrix1.6 Solving the System of Linear Equations 1.6.1 Inversion Method 1.6.2 Gaussian Elimination and Gauss Jordan Elimination 1.6.3 Cramer’s Rule1.7 Matrix Applications 1.7.1 Application of Inversion Method 1.7.2 Application of Gaussian Elimination and Gauss Jordan Elimination 1.7.3 Application of Cramer’s Rule1.8 Input – Output Models

1.1 Introduction

A matrix with m rows ad n columns is said to be of order m by n and this is written m x n matrix.

Matrix can be written as where the first suffix i in element aij

is called the row index of element and the second suffix j is called the column index of element.

Thus we denote an matrix A with element aij by writing

Elements in matrix may be real or complex numbers, or even functions. For example,

i.

ii.

iii.

Example 1.1.1

i) ii)

Definition 1.1A matrix is a rectangular array of elements or entries aij involving m rows and n columns

If then the elements is called the leading diagonal of matrix A

Example 1.1.2

Let

Determinei) Order of matrix Aii) Elements of the leading diagonal of matrix Aiii) Elements

Solution

i) Order of matrix A is ii) 8, 9 and 3 are elements of the leading diagonaliii)

Exercise 1.1.1

Let

Determinei) Order of matrix Aii) Elements of the leading diagonal of matrix Aiii) Elements

Exercise 1.1.2

The matrix H is defined by

i) State that the size of Hii) State

1.2 Types of Matrices

Both D and E are square matrices where

Example 1.2.1

Determine the following matrices are diagonal matrices or not

Solution

Matrix A is not a diagonal matrix because Matrix B is a diagonal matrix because Matrix C is a diagonal matrix

Exercise 1.2.1

Determine the following matrices are diagonal matrices or not?

Definition 1.2

A square matrix is any matrix of order and has the same number of columns and rows

Definition 1.3

A matrix is called a diagonal matrix if

Exercise 1.2.2

Determine the following matrices are scalar matrix or not?

The identity matrix is , where

The identity matrix is , where

Exercise 1.2.3Determine the following matrices are identity matrix or not?

Definition 1.4A matrix is called a scalar matrix if it is a diagonal matrix in which the diagonal elements are equal, that is where k is a scalar

Definition 1.5A matrix is called identity matrix with ones on the main diagonal and zeros everywhere else, that is Identity matrices are denoted by

Exercise 1.2.4Determine the negative matrices of A and B

i) ii)

Example 1.2.2Typical examples of such matrices of order are:

(upper-triangular) (lower-triangular)

Example 1.2.3Determine the transpose of the following matrices

i) ii)

Solution

i) ii)

Definition 1.6

A negative matrix of is denoted by – A where

Definition 1.7

A matrix of is called upper-triangular if every element below the leading

diagonal is zero or for It is called lower-triangular matrix if every element

above the leading diagonal is zero or for

Definition 1.8

If is matrix, then the transpose of A, is the matrix defined

by

Exercise 1.2.5State the transpose of A and B where

i) ii)

Exercise 1.2.6

Given Find then show that the matrix of A is a symmetric matrix?

Exercise 1.2.7

Let find Shows that the matrix of A is a skew symmetric matrix?

Definition 1.9An matrix is called a symmetric matrix if where the elements obey the rule

Definition 1.10An matrix with real entries is called a skew symmetric matrix if where the elements obey the rule so that the leading diagonal must contain zeros

Exercise 1.2.8For the following matrices, determine the matrices to be in row echelon form or not. If the matrices aren’t in row echelon form, give a reason.

i) ii)

iii) iv)

Exercise 1.2.9

For the following matrices, determine the matrices to be in reduced row echelon form or not. If the matrices aren’t in reduced row echelon form, give a reason.

i) ii) iii) iv)

Definition 1.11An matrix A is said to be in row echelon form (REF) if it satisfies the following properties:

i) All zero rows, if thee any, appear at the bottom of the matrixii) The first nonzero entry from the left of a nonzero row is a number 1. This

entry is called a leading ‘1’ of its rowiii) For each nonzero row, the number 1 appears to the right of the leading 1 of

the previous row

Definition 1.12An matrix A is said to be in reduced row echelon form (RREF) if it satisfies the following properties:

i) All zero rows, if thee any, appear at the bottom of the matrixii) The first nonzero entry from the left of a nonzero row is a number 1. This

entry is called a leading ‘1’ of its rowiii) For each nonzero row, the number 1 appears to the right of the leading 1 of

the previous rowiv) If a column contains a leading 1, then all others entries in the column are zero

1.3 Matrix Operations

Example 1.3.1

Given

Findi) A + Bii) A + Ciii) B + C

Solution

i)

ii) A + C (is not possible)iii) B + C (is not possible)

Exercise 1.3.1Given

i)

ii)

Find A + B

Definition 1.13 Let Matrices C = A + B is defined

by Two matrices A and B will be said comfortable for addition only if they are both of

the same order

Exercise 1.3.2

Given

i) B – Cii) B – Aiii) B – C

Properties of Matrices addition and SubtractionIf

i) ii)

iii) iv)

Example 1.3.2

Given

Findi) 3A ii) -2A

Solution

i)

Definition 1.14 If then the A - B is an

Two matrices A and B will be said comformable for subtraction only if they are both of the same order

Definition 1.15Let is an matrix, then the scalar multiplication is denoted

ii)

Exercise 1.3.3

Given

Find

i) 2A ii) -A iii)

Properties of Scalar Multiplication

If

i)

ii)

iii)iv)v)

Exercise 1.3.4

If

i) 2A + 3B ii) A – 3B iii)

Example 1.3.3

Given

Findi) AB ii) BA

Solution

i)

ii) BA (is not possible)

Exercise 1.3.5

Given

Findi) AB ii) BA

Definition 1.16 Suppose A is an matrix and B is a matrix. For the product AB to exist, it

must be that n = p, that is the number of columns in A must be the same as the number of rows in B

Properties of Matrix Multiplication

If A, B and C are matrices, I identity matrix and 0 zero matrix, theni) (distributive law)

ii) (distributive law)

iii) (associative)

iv)

Exercise 1.3.6

Given

Findi) BA ii) ABiii) A2 iv) B2

v) BD vi) DBvii) xi)

1.4 Determinants

Example 1.4.1

Find the determinant of matrix

Solution

Exercise 1.4.1

Given find the determinant of

i) A ii) AT

Definition 1.17

If is a matrix, then the determinant of A denoted by and is

given by

Definition 1.18

Let is a matrix, then the determinant of A given by

Method calculation:

Example 1.4.2

Given

Solution

So that,

Exercise 1.4.2

Given

Note: Methods used to evaluate the determinant above is limited to only and matrices. Matrices with higher order can be solved by using minor and cofactor methods.

Minor, Cofactor and Adjoint

Example 1.4.3

Let Evaluate the following minors

i) M11 ii) M12 iii) M13

Solution

i) ii)

iii)

Example 1.4.4

Let Evaluate the following cofactors

i) C11 ii) C23 iii) C13

Solution

i)

ii)

iii)

Exercise 1.4.3

Definition 1.19 Let

Matrices submatrix of A is obtained by deleting the i-th row and j-th

column of a A denoted by

The minor of is defined as

The cofactor of a is defined as

Let

Calculate the minor and cofactor ofi) M11 and C11 ii) M12 and C12 iii) M13 and C13

Example 1.4.5

Definition 1.20If is cofactor of matrix A, then the determinant of matrix A can be obtained byi) Expanding along the i-th row

OR

ii) Expanding along the j-th column

Find the determinant of by expanding along the first row

Solution

Exercise 1.4.4

Find the determinant of by expanding along the

i) first row ii) second row iii) third row

Properties of Determinant

i) Suppose A is matrix and k a scalar. Suppose the matrix B is obtained by multiplying a single row or column of A by k. Then

ii) If the matrix A is multiplied by k, that is every element in the matrix is multiplied by k, then det

iii) If B is obtaine from A by interchanging two rows or two columns then

iv) Adding or subtraction a multiple of one row (or column) to another row column) leaves the determinant unchanged

v) If A and B are two square matrices such that AB exists then vi) If two rows or two columns of a matrix are equal, the determinant of the matrix is zero

Example 1.4.6

Solution

Definition 1.21Let A is an matrix, then the transpose of the matrix of cofactors of A is called the

matrix adjoint to A, and it is denoted by

We have

Then,

Exercise 1.4.5

Let

1.4 Inverse of Matrix

If and denoted by

Example 1.5.1

Determine matrix is the inverse of matrix

Solution

Note that

Hence

Exercise 1.5.1

Given

Note: If

If

Definition 1.22An matrix A is said to be invertible if there exist an matrix B such that

Theorem 1

If

Example 1.5.2

Let

Solution

Exercise 1.5.2

Find

i) ii) iii) iv)

Example 1.5.3

Let

Solution

From the example 2.23,

Then,

Exercise 1.5.3

Find the inverse of

Theorem 2

If is matrix and

Determination of an Inverse Matrix by Elementary Row Operation

Characteristic of Elementary Row Operations (ERO)

i) Interchange the i-th row and j-th row of a matrix, written as ii) Multiply the i-th row of a matrix by a nonzero scalar k, written as iii) Add or subtract a constant multiple of i-th row to the j-th row, written as

Exercise 1.5.4

Use Elementary Row Operations (ERO) to find the

Theorem 3

Let

If augmented matrix may be reduced and will transform it to by using

elementary row operation (ERO)

1.6 Solving the Systems of Linear Equation

1.6.1 Inversion Method

System of linear equations with m equations and n unknowns

System can be written as

Consider a system of equations written in form, that is where

Where A is a square matrix. Note that A and B are matrices with numerical elements. To find an expression for the unknowns, that is the elements of X. Premultiplying both sides of the equation by the inverse of A, if it exists, to obtain

(1)

The left-hand side can be simplified by nothing that multiplying a matrix by its inverse gives the identity matrix, that is

Multi[lying a matrix by the identity matrix has no effect and so So the left-hand side of equation (1) simplifies to

And so, from (1), Exercise 1.6.1

Solve i) ii)

Using the inverse method

1.6.2 Gaussian Elimination and Gauss-Jordan Elimination

Consider the following system of linear equations with m equations and n unknown

The system of linear equations can be written in the augmented form that is matrix and state the matrix in the following form:

By using elementary row operations (ERO) on this matrix such that the matrix A may reduce in the row echelon form (REF). It is called a Gaussian elimination process.

By using elementary row operations (ERO) on this matrix such that the matrix A may reduce in the reduced row echelon form (RREF). The procedure to reduce a matrix to reduced row echelon form is called Gauss-Jordan elimination

Exercise 1.6.2

Solve the system of linear equations by using Gaussian elimination and Gauss-Jordan eliminationi) ii)

1.6.3 Cramer’s Rule

Step to solve n linear inhomogeneous equations in n variables proceed as follows:

i) Compute the determinant of the coefficient matrix, and if proceed to the next step.

ii) Compute the modified coefficient determinants where is derived from by replacing the ith column of by the inhomogeneous vector .

iii) The solutions are given by for

iv) If the Cramer’s rule cannot be applied. In such a case, either a unique solution to the system does not exist or there is no solution.

Exercise 1.6.3

Theorem 4 The unique solution of the system of linear equation in the two unknowns x and y

here are known constants. To find the values of x and y.

Cramer’s rule is method of obtaining the solution of equations like these as the ratio of two determinants.

Cramer’s rule states

Solve the system of linear equations by using Cramer’s Rule

1.7 Matrix Applications

In this chapter we will solve problem for matrix algebra application. We have two types of application for matrix problem. First we want to solve the business problem using Inversion Method, Gauss Elimination, Gauss Jordan Elimination mehod and Cramer’s Rule. Second application, we want to study the relationship between industrial production and final demand using Input-Output Models.

1.7.1 Application of Inversion Method

Exercise 1.7.1

The management of Checkers Rent-A-Car plans to expand its fleet of rental cars for the next quarter by purchasing compact and full-size cars. The average cost of a compact car is RM 10000, and the average cost of a full-size car is RM 24000.i) If a total of 800 cars are to be purchased with a budget of RM 12 million, how many cars

of each size will be acquired?ii) If the predicted demand calls for a total purchase of 1000 cars with a budget of RM 14

million, how many cars each type will be acquired?

Exercise 1.7.2

Grand Canyon Tours offers air and ground scenic tours of the Grand Canyon. Tickets for the 7.5 hours cost RM169 for an adult and RM129 for a child and each tour group is limited to 19 people. On three recent fully booked tours, total receipts were RM 2931 for the first tour, RM3011 for the second tour and RM2771 for the third tour. Determine how many adults and how many children were in each tour.

1.7.2 Application of Gaussian Elimination and Gauss Jordan Elimination Method

Inversion Method

Exercise 1.7.3

Ahmad inherited RM250000 and invested part of it in a money market account, part in municipal bonds and part in a mutual fund. After one year, he received a total of RM1620 in simple interest from three investments. The money market paid 6% annually, the bonds paid 7% annually and the mutually fund paid 8% annually. There was RM6000 more invested in the bonds than the mutual funds. Find the amount Ahmad invested in each category using Gauss Elimination Method.

Exercise 1.7.4

A total of RM5000 is invested in three funds paying 6%, 8% and 10% simple interest. The yearly interest is RM3700. Twice as much money invested at 6% as invested at 10%. How much was invested in each of the funds.

1.7.3 Application of Cramer’s Rule

Example 1.7.3.1A salesman has the following record of sales during three months for three items A, B

and C which have different rates of commission.

Months Sales of Units Total commission drawnA B C

JanuaryFebruaryMarch

90 100 20 130 50 4060 100 30

800900850

Find out the rates of commission on the items A, B and C. Solve by Cramer’s rule.

1.8 Input-Output Models

Cramer’s Rule The unique solution of the system of linear equation in the two

unknowns and .Here and are known constants. To find the values of and .

Cramer’s rule is method of obtaining the solution of equations like these as the ratio of two determinants.

Cramer’s rule states

and

One of the many important applications of matrix theory to the field of economics is the study of the relationship between industrial production and final demand. At the heart of this analysis is the Input-Output Models pioneered by Wassily Leontief, who awarded a Nobel Prize in economics in 1973 for his contributions to the field.

To illustrate this concept, let’s consider an oversimplified economy consisting of three sectors: agriculture (A), manufacturing (M), and services (S). In general, part of the output of one sector is absorbed by another sector through inter industry purchases, with the excess available to fulfil final (consumer) demands. The relationship governing both intra industrial and inter industrial sales and purchases are conveniently represented by means of an input-output matrix:

The first column (read from top to bottom) tells us that the production of 1 unit of agricultural products requires the consumption of 0.2 unit of agricultural products, 0.2 unit of manufactured goods and 0.1 units of services. The second column tells us that the production of 1 unit of manufactured products requires the consumptions of 0.2 unit of agricultural products, 0.4 unit of manufactured products and 0.2 unit services. Finally, the third column tells us that the production of 1 unit of services requires the consumption of 0.1 units each of agricultural and manufactured products and 0.3 units of services.

Example 1.8.1

Input-Output ModelsIn an Input-Output Model, the matrix equation giving the net output of goods and services needs to satisfy final demand is

Where X is the total output matrix, A is the input-output matrix, and D is the matrix representing final demand.

The solution to this equation is

Which gives the amount of goods and services that must be produced to satisfy final demand.

TKK Corporation, a large conglomerate, has three subsidiaries engaged in producing raw rubber, manufacturing tires and manufacturing other rubber based goods. The production of 1 unit of raw rubber requires the consumption of 0.08 unit of rubber, 0.04 units of tires and 0.02 unit of other rubber based goods. To produce 1 unit of tires requires 0.6 unit of raw rubber, 0.02 unit of tires and 0 units of other rubber based goods. To produce 1 unit of other rubber based goods requires 0.3 unit of raw rubber, 0.01 units of tires and 0.06 unit of other rubber based goods. Markets research indicates that the demand for the following year will be RM 200 million for raw rubber, RM 800 million for tires and RM 120 million for other rubber based products. Find the level of production for each subsidiary in order to satisfy this demand.

Solution:

View the corporation as an economy having three sectors and with an input-output matrix given by

Using equation , the required level of production is given by

where and denote the outputs of raw rubber, tires and other rubber-based goods and where

Now,

and

Therefore,

To fulfill the predicted demand, RM822 million worth of raw rubber, RM854 million worth of tires and RM148 million worth of other rubber-based goods should be produced.

Example 1.8.2

An economy is based on three sectors, agriculture (A), energy (E) and manufacturing (M). Production of a ringgit’s worth of agriculture requires an input of RM 0.20 from the agriculture sector and RM 0.40 from the energy sector. Production of a ringgit’s worth of energy requires an input of RM 0.20 from the energy sector and RM 0.40 from manufacturing sector. Production of a ringgit’s worth of manufacturing requires an input of RM 0.10 from the agriculture sector, RM 0.10 from the energy sector and RM 0.30 from the manufacturing sector. Find the output from each sector that is needed to satisfy a final demand of RM 20 billion for agriculture, RM 10 billion for energy and RM 30 billion for manufacturing.

Example 1.8.3

An economy is based on two industrial sectors, coal and steel. Production of a ringgit’s worth of coal requires an input of RM0.10 from the coal sector and RM 0.20 from the steel sector. Production of a ringgit’s worth of steel requires an input of RM 0.20 from the coal sector and RM 0.40 from steel sector. Find the output for each sector that is needed to satisfy a final demand of RM 20 billion for coal and RM 10 billion for steel.