ABSTRACT
This Project examines matrices and three of its 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.
CHAPTER ONE
GENERAL INTRODUCTION
1.0 BACKGROUND OF THE STUDY
In order to unfold the history of Matrices and Its Applications, the influence of matrices in the mathematical world is spread wide because it providesan importantbaseto many of the principles and practices. It is important that we first determine what matrices is. As such, this definition is not a complete and comprehensive answer, but rather a broad definition loosely wrapping itself around thesubject.
“Matrix” is the Latin word for womb, and it retains that sense in English. It can also mean more generally any place in which something is formed or produced.
The origin of mathematical matrices lies with the study of systems of simultaneous linear equations. An important Chinese text from 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 treatises seventh chapter “too much and not enough”, the concept of a determinant first appears, nearly two milkman before its supposed inventions 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 appears 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 Carh Fredrich Gauss (1777-1855), also known as Gaussian Elimination. (Vitull Marie 2012 )
This project seeks to give an overview of the history of matrices and its 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 Mathematical Art, “which displayed the ability to solve a 3X3 system of equations. (Perotti) [13]. The power and progress of Matrices and its 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-year later, Cramer presented his ideas of solving systems of linear equations based on determinants more than50 years after Leibnitz (Darkwing). Interestingly, Cramer provided no proof for solving an n x n system.
As mentioned before, Gauss’ work initially dealt much with solving linear equations but did not have as much to do with matrices. In order for matrix algebra to develop, a proper notation or method of describing the process was necessary. 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 see, linear algebra has become more relevant since the emergence of calculus, even though its the 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 in order to find a solution. The initial work until this period mainly dealt with the concept of unique solutions and square matrices where the number of equations matched the number of unknowns.
With the turn intothe19th century, Gauss introduced a procedure for solving 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 as well as 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 the concepts of combining, swapping, or multiplying rows with each other in order to eliminate variables from certain equations. After variables are determined, the student is then to 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 thecomposite transformation T2T1 is the product of the matrix for T2 times the matrix of T1”(Tucker, 1993). His work dealing with Matrix multiplication culminated in his theorem, the Cayley-Hamilton Theorem. Simply stated, a square matrixsatisfiesMatrices 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 effectively. Regardless of the technology, though, Gaussian elimination is still the best way to solve a system of linear equations (Tucker,1993).
The influence of matrices and it’s applications in the mathematical world is spread wide because it provides an important base for many principles and practices. Some of the things Matrices is 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 solving energy questions in quantum mechanics. It is also used to create simple everyday household games like Sudoku. It is because of these practical applications thatMatriceshas spread so far and advanced. 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 its application continues to provide the foundation. Even though every few years, companies update their textbooks; the fundamentals stay the same.(laura smoller (2001)[9].
1.2 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 will discuss how the Leontief model represents the economy as a system of linear equations to calculate the gross domestic products and goods production efficiently.
1.3 AIMS AND OBJECTIVES
- To apply matrices to Cryptography, Economic Models and Systems of Linear Equations.
- To improve the methods by which an increase in production output can be achieved.
- To show ways in which sensitive information can be passed across mathematically.
- To disseminate these improved methods to the relevant communities and end-use.
REFERENCES
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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 Wilky and sons, int.
4. Budiman Johannes, (2001) New York: springer,
5. Carlson David: Johnson, Charles R; Lay, David C; porter A. Duane (1993). The linear algebra Curriculum Study Group Recommendations for the first course in linear Algebra. 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
10. Lay, David, C. (2006.) linear algebra and its applications, Third Edition. New York: Pearson,
11. Leontief, Wassily. (2006) Input-output economics, second edition New York Oxford University Press.
12. Linear Algebra: (2000) Determinant, inverses, Rank.pdf
13. Perotti (N.D) (2001) history of linear algebra. Retrieved from http//www.science.unitnith/perotti/history of linear algebra.pdf
14. Sohn, Ira, Eds. (1986) Reading input-output analysis theory and application. New York, oxford university press
15. Strang G. (1993). The fundamental theorem of linear algebra. The America mathematical monthly, 100(9), 848-855
16. Tucker A. (1993). The growing importance of linear algebra in undergraduate mathematics. The college mathematics journal.
17. Vitull:, Marie. (2012) “a brief History of linear Algebra and matrix theory” Department of mathematics. The University of Oregon. Archived from the original retrieved 2014-07-08
18. Weinstein, E.W. (1995) linear algebra, from math world, a wolfram web resource https://mathworld.wolfram.com/LinearAlgebra.html.
Matrices and Its Applications