# Application Of Matrix In Solutions Of Some Physical Problems In Science And Engineering

## Abstract

This research work focuses on the application of matrix in solutions of some physical problems in science and engineering which arise in every human endeavor. This project work attempt to present some physical problems from the selected field (aspect) mentioned, so as to find the solution of the problems. The problems include; using matrice in the process of encrypting and decrypting data in this information age, solving electric circuit problem, balancing a complex chemical equation which is a difficult task for student offering chemistry, modeling of traffic flow to aid free flow, solving problem of forces acting on a truss, and modeling of electric network.

## Table of Content

### Chapter One

#### 1.0 Introduction

- 1.1 Background of Study
- 1.2 Statement of the Problem
- 1.3 Aim and Objectives of the Study
- 1.3.1 Aim of the Study
- 1.3.2 Objectives of the Study
- 1.4 Methodology
- 1.5 Scope and limitation of Study
- 1.5.1 Scope of the Study
- 1.5.2 Limitation of the Study
- 1.6 Significance of the Study
- 1.7 Definition of Terms

### Chapter Two

#### 2.0 Literature Review

- 2.1 Introduction
- 2.2 Direct Methods
- 2.2.1 Gaussian Elimination Method
- 2.2.2 Pivotal Strategy
- 2.2.3 Gauss-Jordan Elimination method
- 2.2.4 LU Factorisation
- 2.2.5 Matrix Inversion
- 2.3 Indirect Method
- 2.3.1 Jacobi Iteration Method
- 2.3.2 Gauss-Seidel Method

### Chapter Three

#### 3.0 Materials and Methods

- 3.1 Introduction
- 3.2 Application of Matrix in Solution of Physical Problem
- 3.3 Engineering
- 3.3.1 Cryptography
- 3.3.2 Civil Engineering
- 3.3.3 Electrical Engineering
- 3.4 Science
- 3.4.1 Balancing a Chemical Equation
- 3.4.2 Modelling of Flow

### Chapter Four

#### 4.0 Results

- 4.1 Introduction
- 4.2 Engineering
- 4.2.1 Cryptography
- 4.2.2 Civil Engineering
- 4.2.3 Electrical Engineering
- 4.3 Science
- 4.3.1 Balancing a Chemical Equation
- 4.3.2 Modeling of Flow

### Chapter Five

#### 5.0 Discussion of Results, Conclusion and Recommendation

- 5.1 Discussion of Results
- 5.2 Conclusion
- 5.3 Recommendation
- Reference

## List of Figure

- 3.1 Forces acting on a truss
- 3.2 Electric Network
- 3.3 Electric Circuit
- 3.4 Traffic Flow

## 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 19th century 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;

A= (?(a_11&a_12&a_13&?&a_1n@a_21&a_22&a_23&?&a_2n@a_31&a_32&a_33&?&a_3n@?&?&?&?&?@a_m1&a_m2&a_m3&?&a_mn ))= (a_ij )

##### Linear System:

By a system of linear equations, we mean an expression of the form:

∑_(i=1)^m??a_ij x_j= b_i (j=1,2,…,n)?

##### Square Matrix:

A matrix involving the same number of rows and columns is called a square matrix, and the number of rows is called its order. The diagonal contained the elements a11, a22, …ann of a square matrix of order n. These elements a11, a22, …,ann are called the principal diagonal.

##### Identity Matrix:

An identity matrix is a unit matrix (denoted by I) with all its diagonal elements equal to 1.

##### Equality of Matrix:

Two or more matrix of the same order are said to be equal if and only if their corresponding entries are equal. Thus, A=B implies aij=bij for all i, j∈N.

##### Triangular Matrix:

This is a square matrix in which all elements below or above the diagonal is zero.

##### Operations of Matrix:

The arithmetic operation of matrice is like the arithmetic of numbers, except that matrice do not necessarily commute under multiplication and matrice need not have multiplicative inverse.

##### Addition of Matrix:

If A= (a_ij )m*n and B= (b_ij )m*n, then A+B= (a_ij+ b_ij )m*n. For all i, j ∈N. Addition is always commutative and associative.

##### Subtraction of Matrix:

Let matrix A be the minuend and B be the subtrahend. Then A-B=A+ (-1)B. In other words, to subtract a matrix, change the sign of the subtrahend (multiplying by-1) and add. For all i, j ∈N. The number being subtracted is called the Subtrahend. The number being subtracted from is called the Minuend.

##### Scalar Multiplication:

A scalar is a real number. Scalar multiplication is the process of multiplying a scalar (number) by a matrix as follows;

∝.A = (?αa?_ij )m*n

##### Matrix Multiplication:

Let A be am×p matrix and B be ap ×n matrix. The product AB is am×n matrix where each element is obtained by multiplying the ith rows and jth column of the product matrix AB. The basic condition for the product of two matrices is that the number of column in matrix A must equal the number of row in matrix B.

##### Transposition:

If A is the general m×n matrix then the n×m matrix obtained from A by interchanging the rows and columns is called the transpose of A, written A^TorA^’.

##### Symmetric:

A matrix A is said to be symmetric if A = AT, that is a_ij=a_ji for all i,j=1,2,…n. However, if AT = -A, then the matrix A is called skewed symmetric, which implies a_ij=?-a?_ji for all i,j=1,2,…n, so in particular all the diagonal elements a_ii are zero.

##### Determinant:

Determinants of a square matrix is nothing but the volume of the (higher dimensional) parallelepiped spanned by the vectors represented by the column (or rows) of matrix. If A is a square matrix, we associate with it a number denoted by |A| or?.The number of ? or |A| is called the determinant of A of order n, written as det (A). Matrix determinant for matrix of order 3 and above is obtained by a process called expansion by cofactor.

##### Minor:

The minor M_ij of a_ij is define as the determinant of the submatrix obtained from matrix A by deleting row i and column j.

##### Cofactor:

The cofactor A_ij of a_ij is define as A_ij=(-1)i+jM_ij. Where M_ijis the minor i.e. the determinant of the sub matrix obtained by deleting row i and column j .

**Note:**

- The expansion by the cofactors of any row or any column will produce the same result.
- The best choice of a row or column about which to expand is one that has the most zero elements.
- The determinant of a matrix with that of its transpose are equal, that is det(A) = det (AT).
- If all the elements of a row (or columns) are zero, then the determinant of that matrix equals zero.
- If any two rows (or columns) are equal or proportional, then the determinant equals zero.
- If we expressed the elements of each row (or column) as the sum of two terms, then the determinants can be expressed as the sum of two determinants having the same order.
- An interchange of any two rows (or columns) changes the signs of the determinant.
- If all elements in any row (or column) are multiplied by a constant, then the determinant is multiplied by the constant.

## Chapter Five

### 5.0 Discussion of Results, Conclusion and Recommendation

#### 5.1 Discussion of Results

The results shown in chapter four is the solution to the problems raised in chapter three; we were able to see the application of matrix in solutions of some physical problem in science and engineering.

In 3.2.2 example 2, we considered using matrix to encrypt and decrypt the phrase “PREPARE TO NEGOTIATE”. We were able to see how the phrase was transformed to a matrix-form, the encoding matrix (square matrix) was assumed and also how the receiver was able to get the decoding matrix (inverse of the encoding matrix) then decrypt the phrase which was encoded.

In 3.2.3 example, we considered using matrix to calculate the forces acting on a truss. We used Statics theory to represent the parameters such as length and degree in a linear system. Then, we were able to solve the linear system using matrix inversion method.

In 3.2.4 example 1, we considered using matrix to model an electric network. We were able to model an electric network using point rule and loop rule. Then we calculated the current flowing through the electric network using Gauss-Jordan elimination method.

In 3.2.5 example 2, we considered using matrix to balance chemical reaction. We were able to see how the coefficient of the chemical equation was represented in a matrix-form (square matrix). Then, we used determinant of a matrix and matrix-inversion method to balance the reaction.

#### 5.2 Conclusion

Most of the physical real world problems therefore can be modelled into matrix form so as to find the solution of the problem. Matrix is an indispensable tool in the field of computing and computer graphics. This research work shed light on a very good application of matrices for student which is the balancing of chemical reaction using matrice and modelling of current flowing in an electric circuit.

Therefore, this research work hopes to serve as a guide and help in solving both simple and complex problems.

#### 5.3 Recommendation

This research work is of vital importance in general, if these methods of solving problems will be used. It will also simplify and ease work as well as minimisation of cost.

Due to the facts that the tools of matrix analysis in real world problem applied on a daily basis, it is recommend that;

- The study of matrix should be integrated into the curriculum of other field beside mathematics. Some field which need knowledge of matrice include biology, chemistry, econometrics, engineering, physics, statistic and a wide variety of other fields.
- Most real world problem should be modelled in a matrix-form so as to ease getting the solution process.

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