A Study on

MATRICES

A Mini Project Report in Quantitative analysis for business decision

Submitted to JNTU,

Kakinada in Partial Fulfilment for the Award of the Degree of

MASTER OF BUSINESS ADMINISTRATION

Submitted BySwapna. K

(Reg. No. 13491E0012).

QIS COLLEGE OF ENGINEERING & TECHNOLOGYAn ISO 9001: 2DEPARTMENT OF MASTER OF BUSINESS

ADMINISTRATION008 Certified Institution and Accredited by NBA

(Affiliated to JNTU, Kakinada and Approved by AICTE)Vengamukkapalem, Pondur Road

ONGOLE –523 272.

Content:

S.NO PARTICULARS PAGE NO1 Abstract 32 Keywords 33 Introduction 34 Definition 35 Review of literature 66 Matrices 4-57 Conclusion 68 Reference 6

ABSTRAC:

Matrices can be examined with a two-stage double-standardization and hierarchical clustering procedure that has been widely applied to other transaction flow tables. An illustration is given, using 1967–1975 citations between 22 mathematical journals. Groups oriented to analysis and to algebra are discerned. Certain journals, such as the Proceedings of the American Mathematical Society, are shown to have broad, no specialized tjes with the other periodicals

KEY WORDS: Matrix;

INTRODUCTION: Matrices, in its plural from called matrices is a mathematical discipline developed to solve system of equations of first degree. ARTHUR CAYLEY (1821-1895), an English mathematician, first studied matrices in 1858, while carrying work with linear transformation of the type: x'=ax+by, y'=cx+dy and JAMES JOSEPH SYLVESTAR (1814-97) another English mathematician gave the name ‘matrix’ to a rectangular arrangement of certain number in some rows and columns. Later two renowned mathematicians JACOBI (1804-1851) contributed to the development to the theory of matrices and determinants.

Almost six years after the invention of matrices, in 1925, Heisenberg (1901-1976) a famous physicist recognised in the algebra of matrices, the important tool he need in the work on quantum mechanics. Today matrices are one of matrices to study the dominance with in a group. Demographers use matrices in the study of births and survivals.

DEFINITION: “A system of m,n numbers(real or complex) arranged in the form of an ordered set of ‘m’ rows each row consisting of an ordered set of n numbers between [ ] or ( ) or || || is called a matrices” (DR.M.V.S.S.N.PRASAD )

Another definition is “A matrices is any array of real numbers (or other suitable entries) arranged in row and column.

Example: A= 1 2 3

5 7 8

In the above example, the horizontal lines of elements are said to constitute, row of the matrix and the vertical lines of elements are said to constituted, columns of the matrix. The rows are counted from top to bottom and the columns from left to right. There are 2 rows and 3 columns in the matrix A and it is called a matrix of the type 2x3 matrix.

Structure of matrix:

Matrix is an arrangement of elements in a particular order which gives it a specific structure. An mxn matrix is constructed as show below.

C1 C2 C3 ...........Cj.........Cn

R1 a11 a12 a13..........a1j.........a1n

R2 a21 a22 a23..........a2j........ a2n

A = . ........ ........ ....... ...........

Ri ai1 ai2 ai3 .........aij...........ain

. ....... ......... ......... ...........

. ....... ........ ............. ...........

Rm am1 am2 am3.........amj........amn

Where each aij is a number real or complex.

Types of matrices:

Rectangular of matrix: A matrix in which in which the number of rows is not equal to the number of columns is called a rectangular matrix.

Example: A = 1 2 3 4

7 8 9 2 2x4

Row matrix: A matrix having only is called a row matrix.

Ex: 5 6 is a row matrix of order 1x2

1 2 3 0 is a row matrix of order 1 x 4

Column matrix: A matrix having only one column is called a column matrix.

Ex: 5

6 is a column matrix of order 2 x1.

Null matrix: if every element of a matrix is zero then it is called a null matrix. A null matrix of the type mxn is denoted by 0.

Ex: 0 = 0 0 0 0

0 0 0 0

0 0 0 0

Square matrix: A matrix in which the number of rows is equal to the number of columns is called a square matrix. The number of rows or columns is being called order of the square matrix.

2 3

6 8

Diagonal matrix: A square matrix, in which all the elements except those in the principal

diagonal are zeros, is called a scalar matrix.

Thus A = 1 0 0

0 2 0 is a diagonal matrix.

0 0 3

Scalar matrix: a square matrix in which the diagonal elements are all equal, and all other elements are zero is called scalar matrix.

Ex: A = 5 0 0

0 5 0

0 0 5 is called scalar matrix.

Unit matrix: a square matrix in which each diagonal elements is equal to the unity and all other elements being zero called a unit matrix or identity matrix. Unity matrix of order n is denoted by I

Ex: I= 1 0

0 1 is unity matrices

Triangular matrix: if all the elements below the principal diagonal of a square matrix are zero. It is called upper triangular matrix

Ex: 1 2 9

0 4 6

0 0 8 is an upper triangular matrix

Ex: 1 0 0

2 4 0

9 8 5 is a lower triangular matrix.

Review of literature:

. ARTHUR CAYLEY (1821-1895), an English mathematician, first studied matrices in 1858, while carrying work with linear transformation of the type: x'=ax+by, y'=cx+dy and JAMES JOSEPH SYLVESTAR (1814-97) another English mathematician gave the name ‘matrix’ to a rectangular arrangement of certain number in some rows and columns. Later two renowned mathematicians JACOBI (1804-1851) contributed to the development to the theory of matrices and determinants.

Conclusion:

I concluded that in this matrix having the rows and columns, horizontal lines are called rows and vertical lines are columns. The matrix are write to the 2x2,2x3,3x3.... and also matrixes having the numbers of types these are column matrix, row matrix, square matrix, unit matrix, triangular matrix, diagonal matrix... etc. These are exam planed above.

Reference:

Dr.M.V.S.S.N.Prasad; quantitative analysis; Radiant publishing house; Hyderabad 2000 Dwivedi, D.N.: quantitative analsis, Vikas Publishing House Pvt. Ltd, New Delhi Mira & Purl: quantitative analysis, Himalaya Publishing House, Mumbai

.