APPLICATION OF MATRICES TO REAL LIFE PROBLEMS.
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APPLICATION OF MATRICES TO REAL LIFE PROBLEMS.
Chapter One: Introduction and Literature Review.
Matrices and determinants were discovered and developed over the 18th and 19th centuries. Initially, their development focused on the transformation of geometric objects and the solution of linear equations.
Historically, the emphasis was on the determinant rather than the matrix. Matrixes are the first topic covered in current Linear Algebra courses.
Matrices offer a theoretical and practical approach to a wide range of problems, including system of linear equation solutions, rigid body equilibrium, graph theory, game theory, the Leontief economics model, forest management, computer graphics and computed tomography, genetics, cryptography, electrical networks, and so on.
Matrices are a valuable tool for expressing and debating challenges that occur from real-life situations. Matrices are used to investigate electrical circuits, quantum physics, and optics, as well as to calculate battery power outputs and resistor conversions of electrical energy into other useful energy.
Matrices play a significant part in the projection of three-dimensional pictures onto a two-dimensional screen, resulting in lifelike motion. Matrices are used in Economics to calculate GDP, which aids in the efficient calculation of goods production.
Matrices are the foundational blocks of robot movement. Robot movements are programmed using matrices in row and column. The inputs for controlling robots are derived from matrices. Scientists in several organisations use matrices to record data from their experiments.
HISTORY OF MATRICES
Matrixes have existed since antiquity, but the term “matrix” was not used to describe the concept until 1850. Matrix is the Latin term for womb, and it retains that meaning in English. It can also refer more broadly to any location where anything is manufactured or produced.
Mathematical matrices originated with the study of simultaneous linear equations. Chiu Chang SuanShu, an ancient mathematical treatise, provides the first documented example of using the matrix technique to solve simultaneous equations.
The notion of determinant appeared approximately two millennia before it was said to have been invented by Japanese mathematician Seki Kowa in 1683 or his German colleague Godfried Leibniz.
Matrixes and determinants originated in the second century BC, with traces dating back to the fourth century BC. However, it wasn’t until the late 17th century that the ideas returned and actual development began.
Matrixes developed from the study of linear equations. The Babylonians began investigating issues that led to simultaneous linear equations, some of which are preserved on clay tablets that have survived.
The Chinese, between 200BC and 100BC, came far closer to matrices than the Babylonians. Indeed, it is safe to conclude that the nine-chapter work on Mathematical Art written during the Han Dynasty provides the first known example of matrix methods. One approach would be to use the Gaussian Elimination method (a method for solving simultaneous linear equations).
This approach was not popular among mathematicians until the nineteenth century. A man named Leibniz investigated Co-efficient systems of quadratic forms for fifty years
which led to the development of matrix theory. Many frequent manipulations of simple matrix theory arose long before matrices became the subject of formal study.
Gauss initially described matrix multiplication (which he considers to be a numerical organisation, since he had not yet encountered the concept of matrix algebra) and the inverse of a matrix in the context of a collection of quadratic form coefficients.
Gauss discovered a system of six linear equations with six unknowns while researching Asteroid Pallas between 1803 and 1809. Gauss provided a systematic approach for solving such equations, which is the Gaussian elimination method on the coefficient matrix. The multiplication theorem was shown and published for the first time in an 1812 publication.
In 1844, Eisenstein denoted linear replacements with a single letter and demonstrated how to add and multiply them like regular numbers. It is reasonable to argue that Eisenstein was the first to consider linear substitutions.
Following Leibniz’ use of the determinant, Cramer introduced his determinant-based formula for solving systems of linear equations in 1750, which is now known as “Cramer’s rule”.
Sylvester coined the term “matrix” in 1850. Sylvester defined a matrix as an oblong arrangement of phrases that resulted in numerous determinants from the square assortment included therein.
In 1853, a man named Cayley was the first to write a note discussing the inverse of a matrix. Cayley defined the matrix algebraically, employing addition, multiplication, scalar multiplication, and inverse operations.
He provided a detailed explanation of the inverse of a matrix. Subtraction came soon after adding, multiplying, and computing inverses with matrices.
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