Matrices and its Applications

Matrices and its Applications – Mathematics Project Topics and Materials

ABSTRACT

This Project examines matrices and three of their applications. Matrix theories were used to solve economic problems, which involve methods by 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

CHAPTER ONE: GENERAL INTRODUCTION

1.0 BACKGROUND OF THE STUDY

1.2 STATEMENT OF PROBLEM

1.3 AIMS AND OBJECTIVES

CHAPTER TWO

2.0 LITERATURE REVIEW

CHAPTER THREE: THEORY OF MATRICES

3.0 THEORY OF MATRICES

3.1.1 DEFINITION AND TYPES OF MATRICES

3.2 ADDITION AND SUBTRACTION OF MATRICES

3.3 PROPERTIES OF MATRIX ADDITION

3.4 SCALAR MULTIPLICATION

3.5 MULTIPLICATION OF MATRICES

3.6 PROPERTIES OF MATRIX MULTIPLICATION

3.7 ELEMENTARY ROW OPERATION

3.8 ECHELON AND ROW-REDUCED ECHELON FORMS OF MATRIX

3.9 DETERMINANT OF MATRIX

3.10 PROPERTIES OF DETERMINANT

3.11 INVERSE OF MATRIX

3.12 PROPERTIES OF INVERSE MATRIX

3.13 A METHOD OF COMPUTING THE INVERSE OF A MATRIX

CHAPTER FOUR: APPLICATIONS OF MATRICES

4.0 INTRODUCTION

4.1 APPLICATION OF MATRICES TO CRYPTOGRAPHY

4.2 APPLICATION TO ECONOMICS

4.2.1 OPEN AND CLOSE ECONOMIC SYSTEM

4.3 APPLICATION OF MATRIX TO SYSTEM OF LINEAR EQUATION

4.4 SOLVING A LINEAR SYSTEM USING (THE ROW REDUCTION) METHOD

CHAPTER FIVE: SUMMARY, CONCLUSIONS

5.1 SUMMARY

5.2 CONCLUSIONS

REFERENCES

CHAPTER ONE

GENERAL INTRODUCTION

BACKGROUND OF THE STUDY

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 for many principles and practices. It is important that we first determine what matrices are. 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, which retains that sense in English. It can also mean any place in which something is formed or produced.

The origin of mathematical matrices lies in the study of systems of simultaneous linear equations. An important Chinese text 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)

In the treatise’s seventh chapter “too much and not enough”, the concept of a determinant first appears nearly two milkmen before its supposed invention by the Japanese mathematician SEKI KOWA in 1683 or his german contemporary GOTTFRIED LEIBNIZ (who is also credited with the invention of differential calculus, separately from but simultaneously with Isaac Newton).

More uses of matrix-like arrangements of numbers appear 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 Cart Fridrich Gauss (1777-1855) also known as Gaussian Elimination.(Vitull Marie 2012 )

This project seeks to give an overview of the history of matrices and their 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 “Nine Chapters of the MathematicalArt,” which displayed the ability to solve a 3X3 system of equations (Perotti). The power and progress of Matrices and their 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, Cramer provided no proof for solving an n x n system.

As mentioned before, Gauss’s work initially dealt much with solving linear equations but did not have as much to do with matrices. A proper notation or method of describing the process was necessary for matrix algebra to develop. 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 linear algebra has become more relevant since the emergence of calculus even though its 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 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.

At the turn of the 19th century, Gauss introduced a procedure to solve a linear equation system. His work dealt mainly with 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 and 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 combining, swapping, or multiplying rows with each other to eliminate variables from certain equations. After variables are determined, the student will use back substitution to help find the remaining unknown variables.

Reviewed 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 defined matrix multiplication as, “the matrix of coefficients for the composite transformation T2T1 is the product of the matrix for T2times 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. 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 effectively. Regardless of the technology Gaussian elimination still proves to be the best way to solve a system of linear equations (Tucker,1993).

The influence of matrices and their applications in the mathematical world is wide because it provides an important base for many principles and practices. Some of the things Matrices are used for are to solve systems of linear format, to find least-square best-fit lines to predict future outcomes or find trends, and 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 everyday household games like sudoku. Matrices have spread so far and advanced because of these practical applications. 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 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 their application continues to provide the foundation. Even though companies update their textbooks every few years, the fundamentals stay the same.(laura smaller (2001)[9].

STATEMENT OF PROBLEM

Due to the great need for security for passing sensitive information from one person to another or from one organization to another through electronic technology, cryptography is needed as a solution to this problem.

Also in economics, this research work is going to discuss how the Leontief model is used to represent the economy as a system of linear equations to calculate the gross domestic products and goods production efficiently.

AIMS AND OBJECTIVES

i. To apply matrices to Cryptography, Economic Models and Systems of Linear Equations.

ii. To improve the methods by which an increase in production out-put can be achieved.

iii. To show ways in which sensitive information can be passed across mathematically.

iv. To disseminate these improved methods to the relevant communities and end-use.

REFERENCES

1. Abraham S.( 1966.) Elementary cryptanalysis a, a mathematical approach. Mathematical Association of America, mathematical library,

2. Alan G. Konhem, (1981)cryptography: a primer New York: Wiley inheritance,

3. Anton, H and C. Rorres. (1994) Elementary linear algebra applications version, 11th edition. 4. John Wiley and sons, int.

4. Budimann Johannas,( 2001) new York: springer,

5. Colson David: Johnson, Charles R; Lay, David C; porter A. Duanne (1993). The linear algebra Curriculum Study Group Recommendations for the first linear algebra course. The college mathematics journal 24(1):41-46

6. Froelich, freeman.(1992) Contemporary precalculus through applications, NCSSM, everyday learning.

7. Jelf Christensen April (2012) find project math 2270 Grant Gustafson University of Utah.

8. Kolman, B.(1999) introductory linear algebra with applications

9. Laura Smoller,( April (2001)) UALR Department of History