Project Abstract

Abstract
Matrices are powerful mathematical tools that have widespread applications in various real-life problems across different fields including engineering, physics, economics, computer science, and many more. This research project aims to explore the practical utilization of matrices in solving real-world problems and to demonstrate their significance in modeling and analyzing complex systems. One of the key areas where matrices find extensive application is in the field of computer graphics. Transformations such as scaling, rotation, and translation of objects in a two-dimensional or three-dimensional space can be efficiently represented and manipulated using matrices. This enables the creation of realistic computer-generated images and animations in video games, movies, virtual reality simulations, and other visual media. In the realm of engineering, matrices play a crucial role in solving systems of linear equations that arise in structural analysis, electrical circuits, control systems, and optimization problems. By representing these systems in matrix form, engineers can apply various techniques such as Gaussian elimination, matrix inversion, and eigenvalue analysis to find solutions and make informed decisions in designing and optimizing complex systems. Furthermore, matrices are extensively used in the field of economics for modeling input-output relationships, analyzing market trends, and optimizing resource allocation. Input-output matrices are employed in economic planning to study the interdependencies between different sectors of an economy and to predict the effects of policy changes or external shocks on production levels and economic growth. In the field of physics, matrices are employed to describe physical phenomena such as quantum mechanics, electromagnetic fields, and fluid dynamics. Quantum mechanics, in particular, relies heavily on the mathematical formalism of matrices to represent the state of quantum systems, calculate probabilities of outcomes, and study the behavior of particles at the subatomic level. Overall, the application of matrices to real-life problems demonstrates their versatility and effectiveness in modeling, analyzing, and solving complex systems across diverse disciplines. By understanding the principles of matrix algebra and mastering computational techniques for manipulating matrices, researchers and practitioners can leverage the power of matrices to gain valuable insights, make informed decisions, and innovate solutions to real-world challenges.

Project Overview

INTRODUCTION AND LITERATURE REVIEW

INTRODUCTION

Matrices and determinants were discovered and developed in the 18th and 19th centuries. Initially, their development dealt with transformation of geometric objects and solution of systems of linear equations. Historically, the early emphasis was on the determinant, not the matrix. In modern treatment of Linear Algebra, matrices are considered first.

Matrices provide a theoretically and practically useful way of approaching many types of problems including; Solutions of system of linear equations, Equilibrium of rigid bodies, Graph theory, Theory of games, Leontief economics model, Forest management, Computer graphics and Computed tomography, Genetics, Cryptography, Electrical networks, etc.

Matrices are a very important tool in expressing and discussing problems which arise from real life issues. Matrices are applied in the study of electrical circuits, quantum mechanics and optics, in the calculation of battery power outputs and resistor conversion of electrical energy into another useful energy.

Matrices play a major role in the projection of three-dimensional images into a two-dimensional screen creating the realistic seeming motion. Matrices are used in calculating the gross domestic products in Economics which eventually helps in calculating the goods production efficiently.

Matrices are the base elements for robot movements. The movements of robots are programmed with the calculation of matrices row and columns. The inputs for controlling robots are given based on the calculations from matrices. Matrices are also used in many organizations by scientists for recording data of their experiment.

HISTORY OF MATRICES

The history of matrices goes back to ancient times, but the term “matrix” was not applied to the concept till 1850. Matrix is the Latin word for womb and it retains that sense in English. It can also mean more generally any place where something is formed or produced.

The origin of mathematical matrices lies with the study of simultaneous linear equations. An important text of the mathematical Art Chiu Chang SuanShu gives the first known example of the use of the matrix method to solve simultaneous equations. The concept of determinant first appeared nearly two millennia before its supposed invention by the Japanese Mathematician Seki Kowa in 1683 or his German contemporary Godfried Leibnitz.

The beginning of matrices and determinants goes back to the 2nd century BC although traces can be seen back to the 4th century BC. However, it was not until near the end of the 17th century that the ideas reappeared and development really got under way. The beginning of matrices arose through the study of systems of linear equations. The Babylonians first started studying problems which led to simultaneous linear equations and some of these are preserved in clay tablets which survived.

The Chinese between 200BC and 100BC came much closer to matrices than the Babylonians. Indeed, it is fair to say that the nine chapters’ text on Mathematical Art written during the Han Dynasty gives the first known example of matrix methods. One method would include what is now known as the Gaussian Elimination method (which is a method used to solve simultaneous linear equations). This method was not popular to mathematicians until the 19th century. The matrix theory was the result of a fifty-year study done by a man named Leibniz who studied Co-efficient systems of quadratic forms. Many common manipulations of the uncomplicated matrix theory appeared long before matrices were the object of mathematical investigation.

Gauss first started to describe matrix multiplication (which he thinks of as an organization of numbers, so he had not yet reached the concept of matrix algebra) and the inverse of a matrix in the particular context of the collection of coefficients of quadratic forms. During Gauss’ work on the study of Asteroid Pallas done between 1803 and 1809, Gauss obtained a system of six linear equations with six unknowns. Gauss gave a systematic method for solving such equations which is precisely Gaussian elimination method on the coefficient matrix. The multiplication theorem was proven and published for the first time in an 1812 paper.

Eisenstein, in 1844, denoted linear substitutions by a single letter and showed how to add and multiply them like ordinary numbers. It is rational to state that Eisenstein was the first to think of linear substitutions. Cramer presented his determinant based formula for solving systems of linear equations which is today known as the “Cramer’s rule” in 1750 after Leibniz’ use of determinant.

The first person to use the term “matrix” was Sylvester in 1850. Sylvester defined a matrix to be an oblong arrangement of terms and saw it as something that led to various determinants from square assortment contained within it. In 1853, a man named Cayley was the first to publish a note which spoke on the inverse of a matrix. Cayley defined the matrix algebraically using addition, multiplication, scalar multiplication and inverses. He gave a precise explanation of an inverse of a matrix. After using addition, multiplication and inverses with matrices, subtraction was soon to follow.