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**CHAPTER ONE**

**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.

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**1.1 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.

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### 1.2 SCOPE OF STUDY

In this study we are going to focus on m x n matrices of different order, i.e 2 x 3, 3 x2, 3 x3, etc. algebra of matrices, i.e. the different operation of addition, subtraction, scalar multiplication, matrix multiplication (under which we will consider power of matrices) and see if division is defined for matrices, determinant of different order starting with 2,3, etc.

Also of square matrix (under which we consider cofactors and adjoint) and the different properties of determinant; inverse of square matrix, product of a square matrix and it’s inverse and also special types of square matrix and the different applications of matrices.

#### 1.3 SIGNIFICANT OF STUDY

** **Matrices are key tools in linear algebra. One of the uses of matrices is to represent linear transformations, which are higher dimensional analogs of linear functions where matrix multiplication corresponds to composition of linear transformation which is used in computer graphics to project 3- dimensional space onto a 2- dimensional screen.

A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations and for a square matrix, the determinant and inverse matrix (when it exists) govern the behaviour of solution of the corresponding system of the linear equations and eigenvalue and eigenvectors provide insight into the geometry of the associated linear transformation. The study of matrix is applicable to every aspect of human endeavour.

1.4 **TYPES OF MATRICES**

1.4.1** Row Matrix**

A row matrix consists of 1 column only e.g. (3 2 4) is a row matrix of order 1 x 3.

1.4.2 **Column Matrix**

A column matrix is matrix having only one column.

e.g. is a column matrix of order 3 x 1

** **So to conserve space in printing, a column matrix is sometimes written on one line with “curly” bracket e.g. (3 2 4) and is the same matrix as order 3 x 1.

1.4.3 **Single Element Matrix **

A single matrix number may be regarded as a matrix as a [x] matrix is having 1 row and 1 column.

**1.4.4 Double Suffix Matrix**

Each element in a matrix has its own particular address or location which can be defined by a system of double suffixes, the first indicating the row and the second the column, thus:

indicates element in the third row and element in the second column.

1.4.5 **Matrix Notation**

A whole matrix can be denoted by a single general element enclosed in brackets, or by a single letter printed in bold types. This is a very neat shorthand and saves much space, for example:

can be denoted by or (a) or by A

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1.5 **SPECIAL MATRICES**

1.5.1 **Square Matrix**

A square matrix is a matrix of order m x m meaning of the same number of rows and columns. e.g.

1 2 5

6 8 9 is a 3 x 3 matrix

1 7 4

A square matrix (a_{ij}) is symmetric of a_{ij} = a_{ji}

1 2 5 1 2 5

2 8 9 = 2 8 9

5 9 4 5 9 4

i.e. it is symmetrical about the leading diagonal.

Note: A = A

A square matrix (aij) is skew-symmetric of a_{ij} = -a_{ji} e.g.

1 2 5 -1 -2 -5

2 8 9 = – -2 -8 -9

5 9 4 -5 -9 -4

in this case A = -A^{T}

**1.5.2 Diagonal Matrix**

A square matrix is called a diagonal matrix, if all its non-diagonal element are zero e.g.

1 0 0

0 3 0

0 0 4

1.5.3 **Unit or Identity Matrix**

A square matrix is called a unit matrix if all the diagonal elements are unity and non-diagonal elements are zero e.g.

(i) 1 0 0

0 1 0

0 0 1

(ii) 1 0

0 1

1.5.4 **Null Matrix or Law Matrix**

Any where in which the elements are zero is called a Zero Matrix or Null Matrix. e.g.

0 0 0

0 0 0

0 0 0

1.5.5 **Equal Matrix**

Two matrixes are said to be equal if:

(i) They are of the same order

(ii) The elements in the corresponding position are equal.

Thus of A = 2 3 B = 2 3

1 4 1 4

A=B

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1.5.6 **Singular Matrix**

If the determinant of a matrix is zero, then the matrix is known as singular matrix e.g.

then A is a singular matrix.

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1.5.7** Triangular Matrix (Echelon Form)**

A square matrix, all of whose element below the leading diagonals are zero is called an upper triangular matrix. A square matrix, all of whose elements above the leading diagonal are zero, is called a lower triangular matrix. e.g.

1 3 2 1 0 0

0 4 1 4 1 0

0 0 0 6 8 5

Upper triangular matrix Lower triangular matrix

1.5.8** Orthogonal Matrix**

A square matrix A is called an orthogonal matrix, if the product of the matrix A and the transpose matrix A^{1} or ‘A’ is a unity matrix e.g.

A.A^{T} = I

If

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1.5.9 **Non-Singular or Invertible Matrix**

A matrix A is called non-singular matrix if its inverse exist.

**How to get the inverse of matrix A:**

To get the inverse of matrix A the following rules must be observed or followed:

- Interchange the two elements on the diagonal.
- Take the negative of the other two elements.
- Multiply the resulting matrix by or equivalently, divide each element by . In case = O, the matrix A is not vertible or non singular.

Expressing the process of inverting matrixes as a rule we do the following:

- We get the minors of the matrix.
- We sign the minors with the rule (-1) i+j to obtain the cofactors.

- We transpose the cofactors

- Multiply the result with the reciprocal of the determinant of the original matrix i.e.

Using matrix

where C_{T} = B Adj A = C^{T}

Consider A =

1.5.10** Conjugate of a Matrix**

Let

then the conjugate of matrix A is Ă

Ă

1.5.11** Idempotent Matrix**

A matrix, such that A^{2} = A is called an idempotent matrix e.g.

1.5.12 **Periodic Matrix**

A matrix A will be called a periodic matrix, if A^{K+1} = A where K is a positive integer. If K is the least positive integer, for which A^{K+1 }= A, then K is said to be periodic of A. if we choose K=1 we get A^{2} = A and we call it to be idempotent matrix.

1.5.13** Nilpotent Matrix**

A matrix will be called a nilpotent matrix, if A^{K} = 0 (null matrix) where K is a positive integer, if however

is the least positive integer of which A^{K} = 0, then K is the index of the nilpotent matrix.

A = ab b^{2} , A^{2} = ab b^{2} ab b^{2} = 0 0 = 0

-a^{2} -ab -a^{2} -ab -a^{2} -ab 0 0

1.5.14** Involuntary Matrix**

A matrix A will be called an involuntary matrix, if A^{2} = I (unit matrix) since I^{2} = I always, it therefore means that a unit matrix is an involuntary matrix.

1.5.15 **Transpose of a Matrix**

In a given matrix A, we interchange the rows, and the corresponding columns, the new matrix obtained is called the transpose of matrix A and is denoted by A^{0} and A^{1 }e.g.

1.5.16** MATRIX **

The transpose of the conjugate matrix A is denoted by

1.5.17** Unitary Matrix**

A square matrix A is said to be unitary of A^{T} A = I

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**Proof:**

Where p is called the conjugate of matrix A.

Then A ^{∂ }A = I