[PDF] ALGEBRA, CALCULUS & SOLID GEOMETRY - MDU

6180_64_07_08_2021_16_55_41_Semester(Algebra,Calculus,SolidGeometry)(BA1007_1).pdf

Paper Code: BA1007-I Semester-I

ALGEBRA, CALCULUS &

SOLID GEOMETRY

Bachelor of

Arts (B.A.)

Three Year Programme

New Scheme of Examination

DIRECTORATE OF DISTANCE EDUCATION

MAHARSHI DAYANAND UNIVERSITY, ROHTAK

(A State University established under Haryana Act No. XXV of 1975)

NAAC 'A+" Grade Accredited University

Copyright © 2003, Maharshi Dayanand University, ROHTAK

All Rights Reserved. No part of

this publication may be reproduced or stored in a retrieval system or transmitted

in any form or by any means; electronic, mechanical, photocopying, recording or otherwise, without the written

permission of the copyright holder.

Maharshi Dayanand University

ROHTAK

- 124 001

Contents

S. No.

Title Page No. 1. MATRICES 1-42 2. SYSTEM OF LINEAR EQUATIONS 43-52 3. EQUATION AND POLYNOMIAL 53-74 4. SOLUTION OF CUBIC AND BIQUADRATIC EQUATIONS 75-86

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UNIT - 1

MMAATTRRIICCEESS

1.0 Introduction

1.1 Objectives

1.2 Review of Matrices

1.2.1 Matrix

1.2.2 Zero Matrix or Null Matrix. 1.2.3 Square matrix. 1.2.4 Row Matrix. 1.2.5 Column Matrix.

1.2.6 Diagonal Matrix.

1.2.7 Scalar Matrix. 1.2.8 Unit Matrix or Identity Matrix. 1.2.9 Triangular matrix. 1.2.10 Sub Matrix 1.2.11 Transpose of a matrix 1.2.12 Conjugate of a Matrix 1.2.13 Transpose Conjugate of a Matrix 1.2.14 Adjoint of a Square Matrix 1.2.15 Inverse of Square Matrix 1.2.16 Singular and Non Singular Matrices 1.2.17 Solution of System of Linear Equations

1.3 Symmetric and Skew-Symmetric Matrices

1.3.1 Symmetric Matrix

1.3.2 Skew Symmetric Matrix

1.4 Hermitian and Skew-Hermitian Matrix

1.4.1 Hremitian Matrix 1.4.2 Skew Hermitian Matrix

1.5 Rank of a Matrix

1.5.

1 Elementary Row (column) Operation on a Matrix

1.5.2 Row Echelon Matrix 1.5.3 Row reduced Echelon Form 1.5.4 Row Rank and Column Rank of a Matrix 1.5.5 Rank of Product of Two Matrices

1.6 Elementary Matrices

1.6.1 Some Theorems on Elementary Matrices

1.7 Inverse of a Matrix

1.8 Linear Dependence and Independence of Row and Column Matrices

1.8.1 Linear Dependence 1.8.2 Linear Independence

1.9 Characteristics Matrix

1.10 Cayley-Hamilton Theorem

1.10.1 Inverse of a Matrix using Cayley-Hamilton Theorem

1.12 Summary

1.13 Key Terms

1.14 Question and Exercises

1.15 Further Reading

2 Matrices

1.0 INTRODUCTION

We have studied about matrices and their properties in the previous classes. Now, we are going to learn

about matrices holding some special properties. In this chapter, we learn about symmetric, skew- symmetric, Hermitian, Skew-Hermitian matrices. We shall also study rank of a matrix, row rank and column rank of a matrix. We shall show that for every matrix its rank, row rank and column rank are all equal.

1.1 OBJ

ECTIVES

After going through this unit you will be able to: Differentiate between Symmetric and Skew- Symmetric matrices. Differentiate between Hermitian and Skeew Herrmitian matrices. Know about the sub-matrix and minor of a matrix Find the rank of any matrix Find the inverse of a matrix Differentiate between linearly dependent and independent vectors Find characteristics roots and corresponding characteristic vectors of a matrix Verify Cayley Hamilton theorem for various matrices and use it to find the inverse of a matrix. Learn important theorems related to characteristic roots and characteristics vectors 1.2 REVIEW OF MATRICES

1.2.1.

Matrix An array of mn numbers arranged in m rows and n columns and bounded by square bracket [ ], brackets ( ) or || || is called m by n matrix and is represented as A = mn2m1min2i1in33231n22221n11211 a...aa....a ....... ....a ....a....a ....... ....a ....aa...aaa....aa = mn2m1m in2i1i n22221 n11211 a....aa ................ a....aa ................ a...aa a...aa ...(1) (1) is known as m n matrix in which there are m rows and n columns. Each member of m n matrix is known as an element of the matrix.

Note:

1. In general, we denote a Matrix by capital letter A = [a ij ], where a ij are elements of Matrix in which its position is in i th row and j th column i.e. first suffix denote row number and second suffix denote column number. 2. The elements a 11 , a 22
,..., a nn in which both suffix are same called diagonal elements, all other elements in which suffix are not same are called non diagonal elements. 3. The line along which the diagonal element lie is called the Principal Diagonal.

Algebra, Calculus & Solid Geometry 3

1.2.2. Zero Matrix or Null Matrix.

A Matrix in which each elements is equal to zero is called a zero matrix or null matrix. e.g., 000 000 or 00 00 00 or 000 000 000 are zero matrix respectively of order 2