Term Paper
On
Matrices and its Application

| Chapter-01: Introduction | 1-3 | 1.1 Background of the Study | 1 | 1.2 Origin of the Study | 2 | 1.3 Objective of the Study | 2 | 1.4 Methodology of the Study | 3 | 1.5 Scope and Limitation of the Study | 3 | Chapter-02: Theoretical Overview | 4-8 | 2.1 Definition of Matrix | 4 | 2.2 Matrix Notation | 4 | 2.3 History of Matrix | 5 | 2.4 Types of Matrix | 6 | 2.4.1 Row Matrix | 6 | 2.4.2 Column Matrix | 6 | 2.4.3 Rectangular Matrix | 6 | 2.4.4 Square Matrix | 6 | 2.4.5 Zero Matrix | 7 | 2.4.6 Upper Triangular Matrix | 7 | 2.4.7 Lower Triangular Matrix | 7 | 2.4.8 Diagonal Matrix | 7 | 2.4.9 Scalar Matrix | 7 | 2.4.10 Identity Matrix | 8 | 2.4.11 Transpose Matrix | 8 | 2.4.12 Regular Matrix | 8 | 2.4.13 Singular Matrix | 8 | Chapter-03: Matrices Operation | 9-15 | 3.1. Properties of matrix operation | 9 | 3.1.1 Properties of subtraction | 9 | 3. 1.2 Properties of Addition | 9 | 3.1.3 Properties of Matrix Multiplication | 10 | 3.1.4 Properties of Scalar Multiplication | 10 | 3.1.5 Properties of the Transpose of a Matrix | 10 | 3.2 Matrix Operation | 11 | 3.2.1 Matrix Equality | 12 | 3.2.2 Matrix Addition | 12 | 3.2.3 Matrix Subtraction | 12 | 3.2.4 Matrix Multiplication | 12 | 3.2.5 Multiplication of Vectors | 14 | 3.3 Inverse of Matrix | 15 | 3.4 Elementary Operations | 15 | Chapter-04: Application of Matrix | 16-21 | 4.1 Application of Matrix | 16 | 4.1.1 Solving Linear Equations | 16 | 4.1.2 Electronics | 16 | 4.1.3 Symmetries and transformations in physics | 17 | 4.1.4 Analysis and geometry | 17 | 4.1.5 Probability theory and statistics | 17 | 4.1.6 Cryptography | 18 | 4.2. Application of Matrices in Real Life | 18 | Chapter-05:Findings and Recommendation | 20-22 | 5.1 Findings | 20 | 5.2 Recommendation | 21 | 5.3 Conclusion | 21 | Reference | 22 |

Executive Summary

Matrices are one of the most powerful tools in Mathematics. We have prepared this report, “Matrices and its Application”, to describe about matrices and its application in our life.
The origins of mathematical matrices lie with the study of systems of simultaneous linear equations. An important Chinese text from between 300 BC and AD 200, Nine Chapters of the Mathematical Art (Chiu Chang Suan Shu ), gives the first known example of the use of matrix methods to solve simultaneous equations.
Matrices have been using widely in various sectors of modern life. Matrices are used in inventory model, electrical networks, and other real life situations. In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The individual items in a matrix are called its elements or entries. Matrix is used quite a bit in advanced statistics.
In our study we have focused various types of matrices in chapter two. The major types of Matrices are Row matrix, Column matrix, Rectangular matrix, Square matrix, Diagonal matrix, Identity matrix, Transpose matrix etc.
In chapter three we have discussed about the matrices properties of addition, subtraction, multiplication, transpose. The study also explores the matrices operations, elementary Matrix operations. This is how matrices work.
The study also covers the application of Matrices in Mathematics and real life in different areas of business and science like budgeting, sales projection, cost estimation etc. Also many physical operations such as magnifications, rotations and reflection through a plane can be represented mathematically by matrices. This mathematical tool is not only used in certain branches of sciences but also in genetics, economics, sociology, modern psychology and in industrial management.
The report ends with some findings of analysis and recommendations regarding its applications. Along with its immeasurable benefits it has some limitation also. Some general limitations of matrices are the followings: * Complicated calculations. * Difficulty in finding Determinant of a 4 * 4 matrix and more. * Time consuming. * Inappropriate and doubtful results. * Lengthy procedure involved. * Tends to create confusion which increases the proportion of mistakes.
A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix decomposition methods simplify computations, both theoretically and practically. Because of some drawbacks matrices are not frequently used like other mathematical methods. For more efficient and effective use, matrices should apply in all possible sectors and should practice more and more.
To be more friendly with matrices and its application: * It can be used for computer programming. * Can be used in business for budgeting, sales projection and cost estimation * Scientist can use a spreadsheet to analyze the result of experience * Can be used to compute industry income tax. * Can be used to analysis production and labor cost in industry. * Can be used in allocation of resources and production scheduling.

CHAPTER-1
INTRODUCTION
CHAPTER-1
INTRODUCTION

1.1 Background of the Study
In mathematics, a matrix (plural matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The individual items in a matrix are called its elements or entries. An example of a matrix with 2 rows and 3 columns is

Matrices of the same size can be added or subtracted element by element. But the rule for matrix multiplication is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. A major application of matrices is to represent linear transformations. Another application of matrices is in the solution of a system of linear equations.
Applications of matrices are found in most scientific fields. In every branch of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics, they are used to study physical phenomena, such as the motion of rigid bodies. In computer graphics, they are used to project a 3-dimensional image onto a 2-dimensional screen.
In probability theory and statistics, stochastic matrices are used to describe sets of probabilities; for instance, they are used within the Page Rank algorithm that ranks the pages in a Google search. Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions.
A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research.
Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory. A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function. 1.2 Origin of the Study
It is an opportunity for the students to acquire an in-depth knowledge through the term paper preparing. The world is rapidly changing with new innovations in almost every discipline. We should use the faster calculation and solution tools to solve the problem of different fields. Matrix is a tool which we can apply in both mathematics and other sciences. This mathematical tool simplifies our work to a great extent when compared with other straight forward method. Some of the merely take advantage of the compact representation of a set of numbers in a matrix. Matrices are a key tool in linear algebra, one uses of matrices is to represent linear transformation Matrices can also keep track of the coefficients in a system of linear equations.
We choose to do the study for the reason that in doing that in doing so. We can solve linear transformations and transition by matrix operation.

1.3 Objective Of The Report
The main objective of education is to acquire knowledge. There are two types of objectives of the report. One is primary objective and the other is Secondary objective.
Primary Objective:
The primary objective of this report is to use the theoretical concepts, gained in the classroom situations, in analyzing real life scenarios. so that it adds value to the knowledge base of us. This is also a partial requirement of the fulfillment of the course.
Secondary Objectives:
The secondary objectives are as follows: * To know the basic concept of matrices * To know the different operation of matrices * To know the historical background of matrices * To know the properties of Matrix operations. * To know the different application of matrices

1.4 Methodology of the Study
The study is based on secondary data. The source of secondary data have been processed and analyzed systematically.
Sources of Secondary data: * Text Books * Class Materials * Different report and research paper * Different websites

1.5 Scope and Limitation of the Study
The study focuses on the basics of matrices and the use of matrices. This paper also emphasizes on the uses of matrix in different field like in science, engineering, accounting, economics, inventory, business etc.
During the completion of this term paper following limitations of the study can be mentioned. * Time frame for the study was very limited. * Lack of available information for making comprehensive study * Lack of experiences has acted as constraints in the way of study * Lack of group study to complete the paper * No primary data are considered
It seems to us that this report is as a study report based on the existing information available on the topic “Matrices and its Application”

CHAPTER- 2
THEORETICAL OVERVIEW

CHAPTER- 2
THEORETICAL OVERVIEW

2.1 Definition of Matrix:

A Matrix is a rectangular array of numbers or other mathematical objects, for which operations such as addition and multiplication are defined. Most commonly, a matrix over a field F is a rectangular array of scalars from F. Most of this article focuses on real and complex matrices, i.e., matrices whose elements are real numbers or complex numbers, respectively. More general types of entries are discussed below. For instance, this is a real matrix:
The numbers, symbols or expressions in the matrix are called its entries or its elements. The horizontal and vertical lines of entries in a matrix are called rows and columns, respectively.

2.2 Matrix Notation
Matrices are commonly written in box brackets:

An alternative notation uses large parentheses instead of box brackets:

The specific of symbolic matrix notation varies widely, with some prevailing trends. Matrices are usually symbolized using upper-case letters (such as A in the examples above), while the corresponding lower-case letters, with two subscript indices (e.g., a11, or a1,1), represent the entries. In addition to using upper-case letters to symbolize matrices, many authors use a special typographical style, commonly boldface upright (non-italic), to further distinguish matrices from other mathematical objects. An alternative notation involves the use of a double-underline with the variable name, with or without boldface style, (e.g., ).
The entry in the i-th row and j-th column of a matrix A is sometimes referred to as the i,j, (i,j), or (i,j)th entry of the matrix, and most commonly denoted as ai,j, or aij. Alternative notations for that entry are A[i,j] or Ai,j. For example, the (1,3) entry of the following matrix A is 5 (also denoted a13, a1,3, A[1,3] or A1,3):

Sometimes, the entries of a matrix can be defined by a formula such as ai,j = f(i, j). For example, each of the entries of the following matrix A is determined by aij = i − j.

In this case, the matrix itself is sometimes defined by that formula, within square brackets or double parenthesis. For example, the matrix above is defined as A = [i-j], or A = ((i-j)). If matrix size is m × n, the above-mentioned formula f(i, j) is valid for any i = 1, ..., m and any j = 1, ..., n. This can be either specified separately or using m × n as a subscript. For instance, the matrix A above is 3 × 4 and can be defined as A = [i − j] (i = 1, 2, 3; j = 1, ..., 4), or A = [i − j]3×4.
Some programming languages utilize doubly subscripted arrays (or arrays of arrays) to represent an m-×-n matrix. Some programming languages start the numbering of array indexes at zero, in which case the entries of an m-by-n matrix are indexed by 0 ≤ i ≤ m − 1 and 0 ≤ j ≤ n – 1. This article follows the more common convention in mathematical writing where enumeration starts from 1.
The set of all m-by-n matrices is denoted (m, n).

2.3 History of Matrix
The origins of mathematical matrices lie with the study of systems of simultaneous linear equations. An important Chinese text from between 300 BC and AD 200, Nine Chapters of the Mathematical Art (Chiu Chang Suan Shu ), gives the first known example of the use of matrix methods to solve simultaneous equations.
In the treatise's seventh chapter, "Too much and not enough," the concept of a determinant first appears, nearly two millennia before its supposed invention by the Japanese mathematician Seki Kowa in 1683 or his German contemporary Gottfried Leibnitz (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 appear in chapter eight, "Methods of rectangular arrays," 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 Carl Friedrich Gauss (1777-1855), also known as Gaussian elimination .
The term "matrix" for such arrangements was introduced in 1850 by James Joseph Sylvester. Sylvester, incidentally, had a (very) brief career at the University of Virginia, which came to an abrupt end after an enraged Sylvester hit a newspaper-reading student with a sword stick and fled the country, believing he had killed the student!
Since their first appearance in ancient China, matrices have remained important mathematical tools. Today, they are used not simply for solving systems of simultaneous linear equations, but also for describing the quantum mechanics of atomic structure, designing computer game graphics , analyzing relationships , and even plotting complicated dance steps ! The elevation of the matrix from mere tool to important mathematical theory owes a lot to the work of female mathematician Olga Taussky Todd (1906-1995), who began by using matrices to analyze vibrations on airplanes during World War II and became the torchbearer for matrix theory.

2.4 Types of Matrices:
The various types of Matrices are the followings.

2.4.1 Row Matrix
A row matrix is formed by a single row.

2.4.2 Column Matrix
A column matrix is formed by a single column.

2.4.3 Rectangular Matrix
A rectangular matrix is formed by a different number of rows and columns, and its dimension is noted as: mxn.

2.4.4 Square Matrix
A square matrix is formed by the same number of rows and columns. The elements of the form aii constitute the principal diagonal. The secondary diagonal is formed by the elements with i+j = n+1.

2.4.5 Zero Matrix
In a zero matrix, all the elements are zeros.

2.4.6 Upper Triangular Matrix
In an upper triangular matrix, the elements located below the diagonal are zeros.

2.4.7 Lower Triangular Matrix
In a lower triangular matrix, the elements above the diagonal are zeros.

2.4.8 Diagonal Matrix
In a diagonal matrix, all the elements above and below the diagonal are zeros.

2.4.9 Scalar Matrix
A scalar matrix is a diagonal matrix in which the diagonal elements are equal.

2.4.10 Identity Matrix
An identity matrix is a diagonal matrix in which the diagonal elements are equal to 1.

2.4.11 Transpose Matrix
Given matrix A, the transpose of matrix A is another matrix where the elements in the columns and rows have switched. In other words, the rows become the columns and the columns become the rows.

2.4.12 Regular Matrix
A regular matrix is a square matrix that has an inverse.
2.4.13 Singular Matrix
A singular matrix is a square matrix that has no inverse.

CHAPTERR 3
Analysis and Discussion

CHAPTERR 3
Analysis and Discussion

3.1. Properties of Matrix Operation
The basic properties of matrix operation are discussed below:
3. 1.1 Properties of Addition
The basic properties of addition for real numbers hold true for matrices.
Let A, B and C be m x n matrices 1. A + B = B + A Commutative 2. A + (B + C) = (A + B) + C Associative 3. There is a unique m x n matrix O with A + O = A Additive identity 4. For any m x n matrix A there is an m x n matrix B
(called -A) with A + B = O Additive inverse

3.1.2 Properties of Subtraction

Two matrices may be subtracted only if they have the same dimension; that is, they must have the same number of rows and columns.

Subtraction is accomplished by subtracting corresponding elements. For example, consider matrix A and matrix B.

1 2 3 5 6 7 A = B = 7 8 9 3 4 5

Both matrices have the same number of rows and columns (2 rows and 3 columns), so they can be subtracted 1-5 2-6 3-7 -4 -4 -4 A-B = = 7-3 8-4 9-5 4 4 4

3.1.3 Properties of Matrix Multiplication
Unlike matrix addition, the properties of multiplication of real numbers do not all generalize to matrices. Matrices rarely commute even if AB and BA are both defined. There often is no multiplicative inverse of a matrix, even if the matrix is a square matrix. There are a few properties of multiplication of real numbers that generalize to matrices. We state them now.
Let A, B and C be matrices of dimensions such that the following are defined. Then 1. A(BC) = (AB)C Associative 2. A(B + C) = AB + AC Distributive 3. (A + B)C = AC + BC Distributive 4. There are unique matrices Im and In with
Im A = A In = A Multiplicative identity

3.1.4 Properties of Scalar Multiplication
Since we can multiply a matrix by a scalar, we can investigate the properties that this multiplication has. All of the properties of multiplication of real numbers generalize. In particular, we have
Let r and s be real numbers and A and B be matrices. Then 1. r(sA) = (rs)A 2. (r + s)A = rA + sA 3. r(A + B) = rA + rB 4. A(rB) = r(AB) = (rA)B 3.1.5 Properties of the Transpose of a Matrix
Recall that the transpose of a matrix is the operation of switching rows and columns. We state the following properties. We proved the first property in the last section.
Let r be a real number and A and B be matrices. Then 1. (AT)T = A 2. (A + B)T = AT + BT 3. (AB)T = BTAT 4. (rA)T = rAT
3.2 Matrix Operation
3.2.1 Matrix Equality
For two matrices to be equal, they must have
1. The same dimensions.
-Each matrix has the same number of rows
-Each matrix has the same number of columns
2. Corresponding elements must be equal. In other words, say that A n x m = [aij] and that B p x q = [bij].
Then A = B if and only if n=p, m=q, and aij=bij for all i and j in range.
Here are two matrices which are not equal even though they have the same elements.

Consider the three matrices shown below.

If A = B then we know that x = 34 and y = 54, since corresponding elements of equal matrices are also equal. We know that matrix C is not equal to A or B, because C has more columns.

Note:
· Two equal matrices are exactly the same.
· If rows are changed into columns and columns into rows, we get a transpose matrix. If the original matrix is A, its transpose is usually denoted by A' or At.
· If two matrices are of the same order (no condition on elements) they are said to be comparable.
· If the given matrix A is of the order m x n, then its transpose will be of the order n x m.

Example 1:
The notation below describes two matrices A and B.

where i= 1, 2, 3 and j = 1, 2

3.2.2 Matrix Addition
If two matrices have the same number of rows and same number of columns, then the matrix sum can be computed:
If A is an MxN matrix, and B is also an MxN matrix, then their sum is an MxN matrix formed by adding corresponding elements of A and B
Here is an example of this: Of course, in most practical situations the elements of the matrices are real numbers with decimal fractions, not the small integers often used in examples.

3.2.3 Matrix Subtraction
If A and B have the same number of rows and columns, then A - B is defined as A + (-B). Usually you think of this as:
To form A - B, from each element of A subtract the corresponding element of B.
Here is a partly finished example:

Notice in particular the elements in the first row of the answer. The way the result was calculated for the elements in row 1 column 2 is sometimes confusing.

3.2.4 Matrix Multiplication
How to multiply two matrices
Matrix multiplication falls into two general categories: * Scalar in which a single number is multiplied with every entry of a matrix * Multiplication of an entire matrix by another entire matrix For the rest of the page, matrix multiplication will refer to this second category.

Scalar Matrix Multiplication
In the scalar variety, every entry is multiplied by a number, called a scalar.

Multiplication a Matrix by a Matrix

It is possible to multiply two matrices if, and only if, the number of columns in the first matrix equals the number of rows in the second matrix.

Otherwise, the product of two matrices is undefined. Matrix C and D below can not be multiplied together because the number of columns in C does not equal the number of rows in D. In this case, the multiplication of these two matrices is not defined.

In order to multiply matrices,
Step 1: Make sure that the number of columns in the 1st one equals the number of rows in the 2nd one. (The pre-requisite to be able to multiply)
Step 2: Multiply the elements of each row of the first matrix by the elements of each column in the second matrix.
Step 3: Add the products.

The matrices can be multiplied since the number of columns in the 1st one, matrix A, equals the number of rows in the 2nd, matrix B. Here the dimension of the product matrix is 4 × 3.

3.2.5 Multiplication of Vectors
This combination of words "multiplication" and "vector" appears in at least four circumstances: 1. multiplication of a vector by a scalar 2. scalar multiplication of vectors 3. multiplication of a vector by a matrix 4. vector multiplication of vectors of which only the fourth may be looked at as a (semi)group operation. Although the rest are also important, here I'll discuss only the latter. The vector multiplication (product) is defined for 3-dimensional vectors. To proceed, we need the notion of right- and left-handedness which apply to three mutually perpendicular vectors.
Two noncollinear (non-parallel) vectors define a plane, and there are two ways to erect a third vector perpendicular to that plane (and, hence, to the two given vectors.) They are distinguished by the right- or left-handed rules. The direction defined by the right-handed rule is customarily preferred to the other one. When one looks from the top of the forefinger (z) the motion from the middle finger (x) towards the thump (y) is positive (counterclockwise).
The late Isaac Isimov once suggested apprehensively that technological advances may lead to thesaurus changes that would eliminate such dear to the heart notions as the clockwise and counterclockwise directions. Luckily, no technological progress could possibly affect the physical underpinning of the right-handed rule.
Obviously the product has no unit element. One the positive side, both the associative and distributive laws hold. For the latter, it's obvious from the geometric considerations. The distributive law implies homogeneity(provided, of course, we first establish some kind of continuity. But this is feasible: small changes in either a or b result in small changes of the area of the parallelogram they define. The plane does not change drastically either.) For a scalar t, where the dot denotes the scalar product and the determinant det(a b c) has vectors a, b, c as its columns. The determinant equals the volume of the parallelepiped formed by the three vectors.

3.3 Inverse of Matrix
The inverse of a square matrix A, denoted by A-1, is the matrix so that the product of A and A-1 is the Identity matrix. The identity matrix that results will be the same size as the matrix A. There is a lot of similarities between real numbers and matrices. For a real number, the inverse of the number was the reciprocal of the number, as long as the number wasn't zero. In Matrices
A(A-1) = I or A-1(A) = I
A-1 does not mean 1/A because there are no Matrix division. A-1 does not mean take the reciprocal of every element in the matrix A.
One of the major uses of inverses is to solve a system of linear equations. It can write a system in matrix form as AX = B. If AX = B, then X = A-1 B
The main reason is because it doesn't always work. * Inverses only exist for square matrices. That means if you don't the same number of equations as variables, then you can't use this method. * Not every square matrix has an inverse. If the coefficient matrix A is singular (has no inverse), then there may be no solution or there may be many solutions, but we can't tell what it is. * Inverses are a pain to find by hand. If you have a calculator, it's not so bad, but remember that calculators don't always give you the answer you're looking for.

3.4 Elementary Matrix Operations
Elementary matrix operations play an important role in many matrix algebra applications, such as finding the inverse of a matrix and solving simultaneous linear equations.
Elementary Operations
There are three kinds of elementary matrix operations. 1. Interchange two rows (or columns). 2. Multiply each element in a row (or column) by a non-zero number. 3. Multiply a row (or column) by a non-zero number and add the result to another row (or column).
When these operations are performed on rows, they are called elementary row operations; and when they are performed on columns, they are called elementary column operations.

CHAPTER- 4
Applications of Matrices

CHAPTER- 4
Applications of Matrices

4.1 Application of Matrix

There are numerous applications of matrices, both in mathematics and other sciences. Some of them merely take advantage of the compact representation of a set of numbers in a matrix. For example, in game theory and economics, the payoff matrix encodes the payoff for two players, depending on which out of a given (finite) set of alternatives the players choose.
4.1.1 Solving Linear Equations

Using matrix methods we can represent a system of linear equations and solve the equations efficiently. Suppose we have a system of equations

This set of equations can be expressed compactly as augmented matrix form as follows

The row operations shown in chapter three perform the basic steps we used to solve systems using elimination on an augmented matrix. This enables us to focus on the numbers without being concerned about algebraic manipulations.

This can be also solved by other method of Matrices. These are more easier to solve than algebraic manipulations.
4.1.2 Electronics

Traditional mesh analysis in electronics leads to a system of linear equations that can be described with a matrix. The behaviour of many electronic components can be described using matrices. Let A be a 2-dimensional vector with the component's input voltage v1 and input current i1 as its elements, and let B be a 2-dimensional vector with the component's output voltage v2 and output current i2 as its elements. Then the behaviour of the electronic component can be described by B = H · A, where H is a 2 x 2 matrix containing one impedance element (h12), one admittance element (h21) and two dimensionless elements (h11 and h22). Calculating a circuit now reduces to multiplying matrices.

4.1.3 Symmetries and transformations in physics

Physicists use a convenient matrix representation known as the Gell-Mann matrices, which are used for the special unitary group SU gauge group that forms the basis of the modern description of strong nuclear interactions, quantum chromodynamics. The Cabibbo–Kobayashi–Maskawa matrix, in turn, expresses the fact that the basic quark states that are important for weak interactions are not the same as, but linearly related to the basic quark states that define particles with specific and distinct masses.

4.1.4 Analysis and geometry
Geometrical optics provides further matrix applications. In this approximative theory, the wave nature of light is neglected. The result is a model in which light rays are indeed geometrical rays. If the deflection of light rays by optical elements is small, the action of a lens or reflective element on a given light ray can be expressed as multiplication of a two-component vector with a two-by-two matrix called ray transfer matrix: the vector's components are the light ray's slope and its distance from the optical axis, while the matrix encodes the properties of the optical element. Actually, there are two kinds of matrices, viz. a refraction matrix describing the refraction at a lens surface, and a translation matrix, describing the translation of the plane of reference to the next refracting surface, where another refraction matrix applies. The optical system, consisting of a combination of lenses and/or reflective elements, is simply described by the matrix resulting from the product of the components' matrices.

4.1.5 Probability theory and statistics
Stochastic matrices are square matrices whose rows are probability vectors, i.e., whose entries are non-negative and sum up to one. Stochastic matrices are used to define Markov chains with finitely many states. A row of the stochastic matrix gives the probability distribution for the next position of some particle currently in the state that corresponds to the row. Properties of the Markov chain like absorbing states, i.e., states that any particle attains eventually, can be read off the eigenvectors of the transition matrices.
Statistics also makes use of matrices in many different forms. Descriptive statistics is concerned with describing data sets, which can often be represented as data matrices, which may then be subjected to dimensionality reduction techniques. The covariance matrix encodes the mutual variance of several random variables. Another technique using matrices are linear least squares, a method that approximates a finite set of pairs (x1, y1), (x2, y2), ..., (xN, yN), by a linear function yi ≈ axi + b, i = 1, ..., N
It can be formulated in terms of matrices, related to the singular value decomposition of matrices. Random matrices are matrices whose entries are random numbers, subject to suitable probability distributions, such as matrix normal distribution. Beyond probability theory, they are applied in domains ranging from number theory to physics.

4.1.6 Cryptography
Cryptography is concerned with keeping communications private. Cryptography mainly consists of Encryption and Decryption. Encryption is the transformation of data into some unreadable form. Its purpose is to ensure privacy by keeping the information hidden from anyone for whom it is not intended, even those who can see the encrypted data. Decryption is the reverse of encryption. It is the transformation of encrypted data back into some intelligible form. Encryption and Decryption require the use of some secret information, usually referred to as a key. Depending on the encryption mechanism used, the same key might be used for both encryption and decryption, while for other mechanisms, the keys used for encryption and decryption might be different.

4.2. Application of Matrices in Real Life
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.
Some of the main applications of matrices are briefed below: * In physics related applications, matrices are applied in the study of electrical circuits, quantum mechanics and optics. * In geology, matrices are used for taking seismic surveys. They are used for plotting graphs, statistics and also to do scientific studies in almost different fields. * Matrices are used in representing the real world data’s like the traits of people’s population, habits, etc. They are best representation methods for plotting the common survey things. * In computer based applications, matrices play a vital role in the projection of three dimensional image into a two dimensional screen, creating the realistic seeming motions. * The matrix calculus is used in the generalization of analytical notions like exponentials and derivatives to their higher dimensions. * In the calculation of battery power outputs, resistor conversion of electrical energy into another useful energy, these matrices play a major role in calculations. Especially in solving the problems using Kirchoff’s laws of voltage and current, the matrices are essential. * For Search Engine Optimization (SEO) Stochastic matrices and Eigen vector solvers are used in the page rank algorithms which are used in the ranking of web pages in Google search. * One of the most important usages of matrices in computer side applications are encryption of message codes. Matrices and their inverse matrices are used for a programmer for coding or encrypting a message. * With the help of matrices internet functions are working and even banks could work with transmission of sensitive and private data’s. * In robotics and automation, matrices are the base elements for the robot movements. The movements of the robots are programmed with the calculation of matrices’ rows and columns. The inputs for controlling robots are given based on the calculations from matrices. * Matrices are used in many organizations such as for scientists for recording the data for their experiments. * Matrices are used in calculating the gross domestic products in economics which eventually helps in calculating the goods production efficiently.

CHAPTER- 5
Findings and
Recommendation

CHAPTER- 5
Findings and
Recommendation

5.1 Findings
Matrices are one of the most powerful tools in Mathematics. The evolution of concept of matrices is the result of a n attempt to obtain compact and simple methods of solving system of linear equation. Matrix notation and operations are used in electronic spreadsheet, programs for personal computer which in turn is used in different areas of business and science like budgeting, sales projection, cost estimation etc. Also many physical operations such as magnifications, rotations and reflection through a plane can be represented mathematically by matrices. This mathematical tool is not only used in certain branches of sciences but also in genetics, economics, sociology, modern psychology and in industrial management. Along with its immeasurable benefits it has some limitation also. Some general limitations of matrices are the followings: * Complicated calculations. * Difficulty in finding DETERMINANT of a 4 * 4 matrix and more. * Time consuming. * Inappropriate and doubtful results. * Lengthy procedure involved. * Tends to create confusion which increases the proportion of mistakes.
Problems with Gauss Elimination * Not quite as easy to remember the procedure for hand solutions. * Round off error may become significant, but can be partially mitigated by using more advanced techniques such as pivoting or scaling.
Problems with Cramer’s Rule * Taking a long time. For 8 equations 2540160 operations, or around 700 hours it requires one operation per second. * Requires a Square system * Round off error may become significant on large problems with non-integer coefficients. * Doesn’t always work if determinant of the coefficient matrix is zero
Problem with Inverse Method * Inverses only exist for square matrices. That means if you don't the same number of equations as variables, then you can't use this method. * Not every square matrix has an inverse. If the coefficient matrix A is singular (has no inverse), then there may be no solution or there may be many solutions, but we can't tell what it is. * Inverses are a pain to find by hand. If you have a calculator, it's not so bad, but remember that calculators don't always give you the answer you're looking for
5.2 Recommendation
A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix decomposition methods simplify computations, both theoretically and practically. Because of some drawbacks matrices are not frequently used like other mathematical methods.
For more efficient and effective use, matrices should apply in all possible sectors and should practice more and more.
To be more friendly with matrices and its application: * It can be used for computer programming. * Can be used in business for budgeting, sales projection and cost estimation * Scientist can use a spreadsheet to analyze the result of experience * Can be used to compute industry income tax. * Can be used to analysis production and labor cost in industry. * Can be used in allocation of resources and production scheduling.
5.2 Conclusion
There are numerous applications of matrices, both in mathematics and other sciences. Some of them merely take advantage of the compact representation of a set of numbers in a matrix. For example, in game theory and economics, the payoff matrix encodes the payoff for two players, depending on which out of a given (finite) set of alternatives the players choose. In addition theoretical knowledge of properties of matrices and their relation to other fields, it is important for practical purposes to perform matrix calculations effectively and precisely. Many problems can be solved by both direct algorithms and iterative approaches. For example, finding eigenvectors can be done by finding a sequence of vectors xn converging to an eigenvector when n tends to infinity. Even matrices are very ancient mathematical concept but it has many applications in our life.