Bsic Mathematics BCS-012 Block-1 Algebra I UNIT-2

Matrices - I
UNIT 2 MATRICES - I
Structure

2.0 Introduction
2.1 Objectives
2.2 Matrices
2.3 Operation on Matrices
2.4 Invertible Matrices
2.5 Systems of Linear Equations
2.6 Answers to Check Your Progress
2.7 Summary


2.0 INTRODUCTION

In this Unit, we shall learn about Matrices. Matrices play central role in
mathematics in general, and algebra in particular. A matrix is a rectangular array
of numbers. There are many situations in mathematics and science which deal
with rectangular arrays of numbers. For example, the following table gives
vitamin contents of three food items in conveniently chosen units.

Vitamin A Vitamin C Vitamin D
Food I 0.4 0.5 0.1
Food II 0.3 0.2 0.5
Food III 0.2 0.5 0
The above information can be expressed as a rectangular array having three rows
and three columns.

0.4 0.5 0.1
0.3 0.2 0.5
0.2 0.5 0
The above arrangement of numbers is a matrix of order 3 3. Matrices have
become an important an powerful tool in mathematics and have found
applications to a very large number of disciplines such as Economics, Physics,
Chemistry and Engineering.

In this Unit, we shall see how Matrices can be combined thought the arithematic
operations of addition, subtraction, and multiplication. The use of Matrices in
solving a system of linear equations will also be studied. In Unit 1 we have
already studied determinant. It must be noted that a matrix is an arrangement of
numbers whereas determinant is number itself. However, we can associate a
determinant to every square matrix i.e., to a matrix in which number of rows is
equal to the number of columns. 29

,Algebra - I
2.1 OBJECTIVES

After studying this Unit, you should be able to :
define the term matrix;
add two or more Matrices;
multiply a matrix by a scalar;
multiply two Matrices;
find the inverse of a square matrix (if it exists); and
use the inverse of a square matrix in solving a system of linear equations.


2.2 MATRICES

We define a matrix as follows :

Def : A m n matrix A is a rectangular array of m n real (or complex
numbers) arranged in m horizontal rows an n vertical columns :



a11 a12 ……… a1j ……… a1n
a21 a22 ……… a2j ……… a2n
: : : :
: : : :
A = ai1 ai2 ……… aij ……… ain ith row
: :
: …(1)
am1 am2 ……… amj ……… amn
jth
column

As it is clear from the above definition, the ith row of A is (aij ai2 … ain)
(1 ≤ i ≤ m) and the jth column is
a1j
a2j
: (1 ≤ j ≤ n)
:
amj

We also note that each element a ij of the matrix has two indices : the row
index i and the column index j. a ij is called the (i,j )th element of the matrix.
For convenience, the Matrices will henceforward be denoted by capital
letters and the elements (also called entries) will be denoted by the
30 corresponding lower case letters.

,The matrix in (1) is often written in one of the following forms : Matrices - I


A = [aij]; A = (aij), A = (a ij )m n or A = (a ij )m n



With i = 1, 2, ………….., m and j = 1, 2, …………, n

The dimension or order of a matrix A is determined by the number of rows
and columns of the matrix. If a matrix A has m rows and n columns we
denote its dimension or order by m n read “m by n”.


For example, A = is a 2 2 matrix and B = is a 2 3
order matrix.

Note a that an m n matrix has mn elements.

Type of Matrices

1. Square Matrix : A square matrix is one in which the number of rows
is equal to the number of columns. For instance,



A= , B= , C=


are square Matrices.

If a square matrix has n rows (and thus n columns), then A is said to be a
square matrix of order n.

2. Diagonal Matrix : A square matrix A[ ] n n for which
diagonal matrix.

For instance,
10 0 0 0
4 0 0
0 0 0 0
D= 0 2 0 and E =
0 0 3 0
0 0 6
0 0 0 5
are diagonal Matrices.

If A = [ ] n n is a square matrix of order n, then the numbers a 11 , a22, ,
… a nn are called diagonal elements, and are said to form the main
diagonal of A. Thus, a square matrix for which every term off the main
diagonal is zero is called a diagonal matrix.
31

, Algebra - I 3. Scalar Matrix : A diagonal matrix A = [aij ] n n for which all the terms
on the main diagonal are equal, that is aij = k for i= j and aij = 0 for i ≠ j
is called a scalar matrix.

For instance


H=


are scalar Matrices.

4. Unit or Identity Matrix : A square matrix A = [a ij] n n is said to be the
unit matrix or identity matrix if

a ij = 0 if i ≠j
1 if i = j

Note that a unit matrix is a scalar matrix with is on the main diagonal.
We denote the unit matrix having n rows (and n columns) by In.
For example,




5. Row Matrix or Column Matrix : A matrix with just one row of
elements is called a row matrix or row vector. While a matrix with just
one column of elements is called a column matrix or column vector.


For instance, A = [2 5 −15] is a row matrix whereas B

is a column matrix.

6. Zero matrix or Null matrix : An m n matrix is called a zero matrix
or null matrix if each of its elements is zero.

We usually denote the zero matrix by Om n



and are example of zero matrices.

Equality of Matrices

Let A = [aij] m×n and B =[ b ij] r×s be two Matrices. We say that A and B
are equals if
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1. m = r, i.e., the number of rows in A equals the number of rows in B.