ABSTRACT
This Project examines matrices and three of its applications. Matrix theories were used to solve economic problems, which involves methods at which goods can be produced efficiently. To encode and also to decode very sensitive information. This project work also goes further to apply matrices to solve a 3 x 3 linear system of equations using row reduction methods.
TABLE OF CONTENT
TITLE PAGE……………………………………………………… i
CERTIFICATION…………………………………………………... ii
DECLARATION………………………………………………………. iii
DEDICATION…………………………………………………………. iv
ACKNOWLEDGMENT………………………………………………. v
ABSTRACT…………………………………………………………….. vi
TABLE OF CONTENT………………….……………………………… Vii-viii
CHAPTER ONE: GENERAL INTRODUCTION
1.0 BACKGROUND OF THE STUDY……………………………………….. 1
1.2 STATEMENT OF PROBLEM……………………………………………. 5
1.3 AIMS AND OBJECTIVES………………………………………………… 5
CHAPTER TWO
2.0 LITERATURE REVIEW………………………………………………….. 6
CHAPTER THREE: THEORY OF MATRICES
3.0 THEORY OF MATRICES…………………………………………………. 8
3.1.1 DEFINITION AND TYPES OF MATRICES…………………………….... 8
3.2 ADDITION AND SUBTRACTION OF MATRICES……………………... 14
3.3 PROPERTIES OF MATRIX ADDITION…………………………………. 15
3.4 SCALAR MULTIPLICATION…………………………………………….. 15
3.5 MULTIPLICATION OF MATRICES……………………………………… 15
3.6 PROPERTIES OF MATRIX MULTIPLICATION………………………….. 16
3.7 ELEMENTARY ROW OPERATION ………………………………………. 16
3.8 ECHELON AND ROW-REDUCED ECHELON FORMS OF MATRIX…… 17
3.9 DETERMINANT OF MATRIX……………………………………………… 21
3.10 PROPERTIES OF DETERMINANT…………………………………………. 22
3.11 INVERSE OF MATRIX……………………………………………………….. 24
3.12 PROPERTIES OF INVERSE MATRIX………………………………………. 25
3.13 A METHOD OF COMPUTING THE INVERSE OF A MATRIX…………… 25
CHAPTER FOUR: APPLICATIONS OF MATRICES
4.0 INTRODUCTION……………………………………………………………. 25
4.1 APPLICATION OF MATRICES TO CRYPTOGRAPHY…………………… 25
4.2 APPLICATION TO ECONOMICS…………………………………………… 32
4.2.1 OPEN AND CLOSE ECONOMIC SYSTEM………………………………… 32
4.3 APPLICATION OF MATRIX TO SYSTEM OF LINEAR EQUATION………... 40
4.4 SOLVING A LINEAR SYSTEM USING (ROW REDUCTION) METHOD……………………………………………………………….………… 41
CHAPTER FIVE: SUMMARY ,CONCLUSIONS
5.1 SUMMARY……………………………………………………………………….. 46
5.2 CONCLUSIONS……………………………………………………….…………. 46
REFERENC
CHAPTER ONE
GENERAL INTRODUCTION
1.0 BACKGROUND OF THE STUDY
In order to unfold the history of Matrices and Its Applications, the influence of matrices in the mathematical world is spread wide because it provides an important base to many of the principles and practices. It is important that we first determine what matrices is. As such, this definition is not a complete and comprehensive answer, but rather a broad definition loosely wrapping itself around the subject.
“Matrix” is the Latin word for womb, and it retains that sense in English. It can also mean more generally any place in which something is formed or produced.
The origin of mathematical matrices lies with the study of systems of simultaneous linear equations. An important Chinese text from between 300Bc and Ad 200, nine chapters of the mathematical art, gives the first known example of the use of matrix methods to solve simultaneous equations.(Laura Smoller (2012))[9]
In the treatises seventh chapter “too much and not enough”, the concept of a determinant first appears, nearly two milknna before its supposed inventions by the japanese mathematician SEKI KOWA in 1683 or his german contemporary GOTTFRIED LEIBNIZ (who is also credited wiith the invention of differential calculus, seperately from but simutaneously with isaac Newton).
More uses of matrix-like arrangements of numbers appears in which a method is given for solving simultaneous equations using a counting board that is mathematically identical to the modern matrix method of solution outlined by Carh Fridrich Gauss (1777-1855) also known as Gaussan Elimination. (Vitull marie 2012 )[17]
This project seeks to give an overview of the history of matrices and its practical applications touching on the various topics used in concordance with it.
Around 4000 years ago, the people of Babylon knew how to solve a simple 2X2 system of linear equations with two unknowns. Around 200 BC, the Chinese published that “Nine Chapters of the Mathematical Art,” they displayed the ability to solve a 3X3 system of equations. (perotti)[13]. The power and progress in Matrices and its application did not come to fruition until the late 17th century.
The emergence of the subject came from determinants, values connected to a square matrix, studied by the founder of calculus, Leibnitz, in the late 17th century. Lagrange came out with his work regarding Lagrange multipliers, a way to “characterize the maxima and minima multivariate functions.” (Darkwing) More than fifty years later, Cramer presented his ideas of solving systems of linear equations based on determinants more than 50 years after Leibnitz (Darkwing). Interestingly enough, Cramer provided no proof for solving an n x n system.
As mentioned before, Gauss work dealt much with solving linear equations themselves initially, but did not have as much to do with matrices. In order for matrix algebra to develop, a proper notation or method of describing the process was necessary. Also vital to this process was a definition of matrix multiplication and the facets involving it. “The introduction of matrix notation and the invention of the word matrix were motivated by attempts to develop the right algebraic language for studying determinants. In 1848, J.J. Sylvester introduced the term “matrix,” the Latin word for womb, as a name for an array of numbers. He used womb, because see, linear algebra has become more relevant since the emergence of calculus even though it’s foundational equation of ax+ b=0 dates back centuries.
Euler brought to light the idea that a system of equations doesn’t necessarily have to have a solution. He recognized the need for conditions to be placed upon unknown variables in order to find a solution. The initial work up until this period mainly dealt with the concept of unique solutions and square matrices where the number of equations matched the number of unknowns.
With the turn into the 19th century Gauss introduced a procedure to be used for solving a system of linear equations. His work dealt mainly with the linear equations and had yet to bring in the idea of matrices or their notations. His efforts dealt with equations of differing numbers and variables as well as the traditional pre-19th century works of Euler, Leibnitz, and Cramer. Gauss’ work is now summed up in the term Gaussian elimination. This method uses the concepts of combining, swapping, or multiplying rows with each other in order to eliminate variables from certain equations. After variables are determined, the student is then to use back substitution to help find the remaining unknown variables.
He viewed a matrix as a generator of determinants (Tucker, 1993). The other part, matrix multiplication or matrix algebra came from the work of Arthur Cayley in 1855.
Cayley’s defined matrix multiplication as, “the matrix of coefficients for the composite transformation T2T1 is the product of the matrix for T2 times the matrix of T1” (Tucker, 1993). His work dealing with Matrix multiplication culminated in his theorem, the Cayley-Hamilton Theorem. Simply stated, a square matrix satisfies Matrices at the end of the 19th century were heavily connected with Physics issues and for mathematicians, more attention was given to vectors as they proved to be basic mathematical elements. With the advancement of technology using the methods of Cayley, Gauss, Leibnitz, Euler, and others determinants and linear algebra moved forward more quickly and more effective. Regardless of the technology though Gaussian elimination still proves to be the best way known to solve a system of linear equations (Tucker, 1993).
The influence of matrices and it’s applications in the mathematical world is spread wide because it provides an important base to many of the principles and practices. Some of the things Matrices is used for are to solve systems of linear format, to find least-square best fit lines to predict future outcomes or find trends, to encode and decode messages. Other more broad topics that it is used for are to solve questions of energy in Quantum mechanics. It is also used to create simple every day household games like Sudoku. It is because of these practical applications that Matrices has spread so far and advanced. The key, however, is to understand that the history of linear algebra provides the basis for these applications.
Although linear algebra is a fairly new subject when compared to other mathematical practices, its uses are widespread. With the efforts of calculus savvy Leibnitz the concept of using systems of linear equations to solve unknowns was formalized. Other efforts from Scholars like Cayley. Euler, Sylvester, and others changed matrices into the use of linear algebra to represent them. Gauss brought his theory to solve systems of equations proving to be the most effective basis for solving unknowns.
Technology continues to push the use further and further, but the history of matrices and its application continues to provide the foundation. Even though every few years companies update their textbooks, the fundamentals stay the same.(laura smoller (2001)[9].
1.2 STATEMENT OF PROBLEM
Due to the great need of security for passing sensitive information from one person to another or from one organization to another through electronic technology, there is need for cryptography as a solution to this problem.
Also in economics this research work is going to discuss how Leontief model is used to represent the economy as a system of linear equation so as to calculate the gross domestic products and goods production efficiently.
1.3 AIMS AND OBJECTIVES
i. To apply matrices to Cryptography, Economic Models and system of Linear Equations
ii. To improve the methods at which increase in production out-put can be achieved
iii. To show ways at which sensitive information can be passed across mathematically.
iv. To disseminate this improved methods to the relevant communities and end use
