engineering mathematics 1st year

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1
MATRICES

UNIT STRUCTURE
1.0 Objectives
1.1 Introduction
1.2 Definitions
1.3 Illustrative examples
1.4 Rank of matrix
1.5 Canonical form or Normal form
1.6 Normal form PAQ
1.7 Let Us Sum Up
1.8 Unit End Exercise

1.0 OBJECTIVES

In this chapter a student has to learn the
 Concept of adjoint of a matrix.
 Inverse of a matrix.
 Rank of a matrix and methods finding these.

1.1 INTRODUCTION

At higher secondary level, we have studied the definition of a matrix,
operations on the matrices, types of matrices inverse of a matrix etc.

In this chapter, we are studying adjoint method of finding the inverse of a
square matrix and also the rank of a matrix.

1.2 DEFINITIONS

1) Definitions:- A system of m  n numbers arranged in the form of
an ordered set of m horizontal lines called rows & n vertical lines called
columns is called an m  n matrix.

The matrix of order m  n is written as

, 2

 a11 a12 a13 a1j a1n 
a 
 21 a 22 a 23 a 2 j a 2n 
....... ..... ....... ..... ..... 
 
 a i1 a i2 a i3 a ij a in 
 a m1 a m2 a m3 a mj a mn 
  n n

Note:

i) Matrices are generally denoted by capital letters.
ii) The elements are generally denoted by corresponding small letters.

Types of Matrices:

1) Rectangular matrix :-

Any mxn Matrix where m  n is called rectangular matrix.

For e.g

2 3 4
1 2 3 23

2) Column Matrix :

It is a matrix in which there is only one column.

1 
x   2 
 4  31

3) Row Matrix :

It is a matrix in which there is only one row.

x  5 7 913

4) Square Matrix :

It is a matrix in which number of rows equals the number of
columns.

i.e its order is n x n.

, 3

e.g.

 2 3
A 
 4 6  22

5) Diagonal Matrix:

It is a square matrix in which all non-diagonal elements are zero.

e.g.
2 0 0
A   0 1 0 
 0 0 0  33


6) Scalar Matrix:

It is a square diagonal matrix in which all diagonal elements are equal.

e.g.

5 0 0
A  0 5 0 
0 0 5  33


7) Unit Matrix:

It is a scalar matrix with diagonal elements as unity.

e.g.

1 0 0 
A  0 1 0 
0 0 1  33

8) Upper Triangular Matrix:

It is a square matrix in which all the elements below the principle diagonal
are zero.

, 4

e.g.

1 3 0 
A  0 0 1 
0 0 5  33

9) Lower Triangular Matrix:

It is a square matrix in which all the elements above the principle
diagonal are zero.

e.g.
 0 0 0
A   3 4 0 
 1 3 2  33

10) Transpose of Matrix:

It is a matrix obtained by interchanging rows into columns or columns into
rows.

1 3 5 
A  
3 7 9  23

1 3 
A  Transpose of A  3 7 
T


5 9  32

11) Symmetric Matrix:

If for a square matrix A, A  AT then A is symmetric

1 3 5 
A  3 4 1 
5 1 9 

12) Skew Symmeric Matrix :

If for a square matrix A, A  AT then it is skew -symmetric matrix.