Application of Matrices to Real-Life Problems
Chapter One
Preamble of the Study
The introduction and development of the notion of a matrix and the subject of linear algebra followed the development of determinants, which arose from the study of coefficients of systems of linear equations. Leibnitz, one of the founders of calculus, used determinants in 1963 and Cramer presented his determinant-based formula for solving systems of linear equations (today known as Cramer’s rule) in 1750.
The first implicit use of matrices occurred in Lagrange’s work on bilinear form in late 1700. Lagrange desired to characterize the maxima and minima of multi-variant functions. His method is now known as the method of Lagrange multipliers. To do this he first required the first-order partial derivation to be 0 and additionally required that a condition on the matrix of second-order partial derivatives holds; this condition is today called positive or negative definiteness, although Lagrange did not use matrices explicitly.
CHAPTER TWO
LAWS AND ALGEBRAS OF MATRICES
Introduction
This chapter discusses the different laws of matrices. These are laws of addition, subtraction, multiplication, scalar multiplication, matrix multiplication and properties of matrices.
Addition of Matrices
If A = [ aij ] and B = (bij) be two matrices of same order, m x n matrices. Then their sum A+B is defined as the matrix, each element of which in the sum of the corresponding elements of A and B.
Adding both matrices, we have
If A = [aij] and B = [bij]
then A x B = [aij +[bij]
Addition of Matrices
Only matrices of same order can be added or subtracted.
Let A, B, C be arbitrary matrices of the same order. Then using the definition of sum of matrices and law of addition of real numbers or complex numbers.
Commutative Law:
A + B = [ aij ] + [ bij ] = ( bij + aij ) = B + A.
Associative Law:
A + ( B + C)
= [ aij + (bij + cij ) ] = [ (aij + bij ) + cij ]
= [ ( A + B) + C ]
Proof of (1)
Let A and B be the arbitrary matrices of the same order. Then to show that it is commutative
A + B = (aij ) + ( bij )
= (bij) + (aij) = B + A as required and the proof is complete.
Proof of (2) Let A, B and C be arbitrary matrices of the same order, Then to show associatively, we have A + (B+C) = (aij) +
[( bij + cij)] = (aij + bij) + Cij = (A + B) + C.
CHAPTER THREE
Each n-square matrix A = (aj) is assigned a special scalar called the determinant of A denoted by (A) or /A/ or in the case where the matrix entries are written out in full, the determinant in denoted by surrounding the matrix entries by vertical bars instead the brackets or parentheses of the matrix. For instance the determinant of a matrix
a11 a12 ain
is written
Although most often used for matrices whose entries are real or complex numbers, the definition of determinant only involves addition, subtraction and multiplication and so it can be defined for square matrix with entries taken from any commutative ring. This for instance the determinant of matrix with integer coefficient will be an integer and the matrix has an inverse with integer coefficient of and only if this determinant is 1 or -1.
CHAPTER FOUR
INVERSE OF A MATRIX SQUARE
INVERSE OF A MATRIX
If A and B are two square matrices of the same order such that:
AB = BA = I
Then B is called the inverse of A i.e B=A-1 and A is the inverse of B.
The condition for a square matrix A to posses an inverse in that matrix A is non-singular i.e |A| ≠ 0
CHAPTER FIVE
SUMMARY, CONCLUSION AND RECOMMENDATION
Introduction
It is important to reiterate that the objective of this study was focused on matrix and its everyday application to everyday life and to different areas of study. In the preceding chapters, we have examined the different types of matrices its branches and mathematical application. In this chapter, our goal is to summarize the study and draw a useful conclusion regarding its objectives.
Summary
This study was undertaken to examine matrix and its application to everyday life. The study opened with chapter one where the scope of study, significant of study, types of matrices were defined. The study reviewed related and relevant literatures. The chapter two discussed the different laws of matrices. These are laws of addition, subtraction, multiplication, scalar multiplication, matrix multiplication and properties of matrices. The third chapter described the determinant of various orders. While the fourth chapter highlighted possible ways by which Matrices are applied to real life situations.
CONCLUSION
The term matrix was introduced by the 19th-century English mathematician James Sylvester, but it was his friend the mathematician Arthur Cayley who developed the algebraic aspect of matrices in two papers in the 1850s. Matrices are simply the rectangular arrangement of numbers, expressions, symbols which are arranged in columns and rows. The numbers present in the matrix are called as entities or entries. A matrix is said to be having ‘m’ number of rows and ‘n’ number of columns. They are used for plotting graphs, statistics and also to do scientific studies and research in almost different fields. Matrices are also used in representing the real world data’s like the population of people, infant mortality rate, etc. They are best representation methods for plotting surveys.
A matrix organizational structure is a company structure in which the reporting relationships are set up as a grid, or matrix, rather than in the traditional hierarchy. In other words, employees have dual reporting relationships – generally to both a functional manager and a product manager. Matrix mathematics has many applications. Mathematicians, scientists and engineers represent groups of equations as matrices; then they have a systematic way of doing the math. Some properties of matrix mathematics are important in math theory. Matrices find many applications in scientific fields and apply to practical real life problems as well, thus making an indispensable concept for solving many practical problems and this is what study was poised on revealing.
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