Full Project – APPLICATION OF MATRIX IN SOLUTIONS OF SOME PHYSICAL PROBLEMS IN SCIENCE AND ENGINEERING

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CHAPTER ONE

1.0 INTRODUCTION

1.1 Background of Study

Mathematics is being used in general in almost all aspect of human life. Mathematicians seek out patterns and formulate new conjectures. Therefore mathematical models and method have been used in different areas, such as science, technology, social and management science and arts.

Several problems can be transformed into mathematical form, and its application in the real world cannot be over emphasized. Mathematical modeling in general is been used in areas such as systems of linear differential equation, interpolation polynomials, to describe the behavior of a system, population, linear programming and systems of linear equations.

A huge amount of computer time is spent for carrying out matrix computations for solving linear systems, which is the backbone of many science and engineering problems. The connection between digital computation and matrices is obvious. Since many engineering problems are linear, and since digital computers with finite memory can manipulate only finite sets of numbers, the solution of linear problems by digital computation usually involves matrix.

Almost every physical situation can be captured by the help of a mathematical equation (whether differential equation or a linear equation). If physical problems are transformed into mathematical model with help of differential equation, it may be solved by any appropriate method depending on the nature of the differential equation. Something similar (but somewhat slightly different) happen when a physical problem is translated into a system of linear equations. That is, it can be solved by any of the available methods of solving system of linear equations. We observed here that the field of applied mathematics was generally introduced during 19^thcentury by some famous mathematicians.

This project work did not mean to present all the methods of solving a system of linear equation, but it aim at presenting some selected method called matrix methods of solving system of linear equations the linear equations will be transformed into matrix and the solution can be obtained with the used of some properties of matrix.In the project work therefore we are going to consider some specific areas where matrix are applied, they include solutions of problems in science, and engineering though its application covers other aspect such as social science, management science and financial areas.

The work is divided into five chapters as follows:

Chapter one is an introductory part that contains basic definitions and operations of the tools used in the entire project work.

Chapter two is the literature review. We discuss two method of solving systems of linear equations using matrices: viz direct (exact) method and indirect (iterated) method.

In chapter three, we present a number of various areas where matrix can be applied under the field of science and engineering.

In chapter four, we try to discuss the result of our matrix application obtained in chapter three using various matrix method of solution.

Chapter five comprises of the discussion of results, conclusion and recommendations.

1.2 Statement of the Problem

Problem refers to question decided to be solved upon with an improve procedure (matrix). Each of the areas given consideration in this project has it own problem. In cryptography, there is the problem of using video cipher to encrypt data, which is obsolete today, so we try using matrix to encrypt and decrypt data in this project. In balancing of chemical reaction, since there is no standard rule in balancing a chemical reaction, it is always difficult balancing a complex chemical reaction, so matrix is used in this project to balance a complex chemical reaction.

1.3 Aim and Objectives of the Study

1.3.1 Aim of the Study

The aim of this study is to see how matrixis being used as a focal point in solving physical real world problem.

1.3.2 Objectives of the Study

After solving some selected physical real world problem, we hope to have shown that matrix is an indispensable tool in the application of mathematics (matrix) to solve science and engineering problem.

1.4 Methodology

The methodology involved in this project is to pick out some various physical problems in the fields of science, engineering and then find their solution using matrix method of solution. In the field of Engineering; problems of cryptography, civil engineering and electrical engineering were considered. While in the field of science; problems of balancing chemical reaction and modeling of traffic flow were considered.

Most of the problems in this project were represented mostly as a square matrix. Matrix operation of determinant was used to find most solution to the problems deal with in this project. Expansion by cofactor was the main process used in finding determinant as most of the matrix was of order three and above. Matrix of higher order (e.g Order 7) were solved using matrix application (i.e. Online Matrix Calculator). Matrix inversion is another main method that was used in finding solution to the problems dealt with in this project. No indirect method (i.e. Gausss Seidel and Jacobi iteration method) of solving matrix was used in this project as the project solved in this project is not large enough to require indirect method of solving matrix. Indirect methods are mostly used in the transformation of image from 2D to 3D, genetic engineering and studying the trend of stock exchange which were not considered in this project.

In using matrix to find some solution of physical real world problem; problem of other discipline were useful. In using matrix to model current flowing through an electric circuit and electric network, knowledge of loop rule, point rule, Ohm’s law and Kirchhoff’s law were applied. In civil engineering, Static law which deals with the resolution of forces acting on a truss was used in this project. Kirchhoff’s law was used in the modeling of traffic flow.

1.5 Scope and Limitation of the Study

1.5.1 Scope of the Study

The project is intended to cover only the field of science and engineering. In the field of science, consideration is given to areas like balancing of chemical reaction and modeling of flow. While in the field of engineering, consideration is given to areas like forces acting on a truss, cryptography and modeling of current flowing in an electric circuit.

1.5.2 Limitation of the Study

The project is limited to the field of science and engineering and also limiting the number of areas to cryptography, balancing of chemical reaction, modeling of flow, modeling of current in an electric circuit and forces on a truss.

1.6 Significance of the Study

The significance of this Study cannot be over emphasizes. The study try to show that matrix is an indispensable in the application of Mathematics to solving scientific and engineering problems. The significant of this Study varies based on the point of view, it is being looked from. The use of matrix in cryptography is of great significance to both individual, government and any researchers as it help to keep some information private. The use of matrix in balancing chemical reaction is of benefit to student and chemist researchers. Balancing of chemical equation has no definite patterns, so using matrix to balance it can reduce the stress involved. The use of matrix in modeling traffic flow is significance to traffic flow control agency and individual as it helps the individual against running into deadlock in the traffic. While matrix assists the traffic control agency in ensuring that there is free flow of traffic to avoid accidents.

1.7 Definition of Terms

SCIENCE: Science (from Latin Scientia meaning “knowlegde”) is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe.

ENGINEERING: According to wikipaedia, engineering is the discipline, skill and profession of acquiring and applying scientific, economic, social and practical knowledge in order to design and build structures, machines, devices, systems, materials and processes.

MATRIX: A matrix is a rectangular array of elements consisting of m rows and n columns. The element may be variables or numerals or both. Matrice is the plural form of matrix. Matrix, as a rectangular array of entries is denoted by;

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