56 MATHEMATICS
Chapter 3
MATRICES
d
he
The essence of Mathematics lies in its freedom. — CANTOR
3.1 Introduction
is
The knowledge of matrices is necessary in various branches of mathematics. Matrices
are one of the most powerful tools in mathematics. This mathematical tool simplifies
bl
our work to a great extent when compared with other straight forward methods. The
evolution of concept of matrices is the result of an attempt to obtain compact and
simple methods of solving system of linear equations. Matrices are not only used as a
pu
representation of the coefficients in system of linear equations, but utility of matrices
far exceeds that use. Matrix notation and operations are used in electronic spreadsheet
be T
programs for personal computer, which in turn is used in different areas of business
re
and science like budgeting, sales projection, cost estimation, analysing the results of an
o R
experiment etc. Also, many physical operations such as magnification, rotation and
reflection through a plane can be represented mathematically by matrices. Matrices
tt E
are also used in cryptography. This mathematical tool is not only used in certain branches
of sciences, but also in genetics, economics, sociology, modern psychology and industrial
C
management.
In this chapter, we shall find it interesting to become acquainted with the
no N
fundamentals of matrix and matrix algebra.
3.2 Matrix
Suppose we wish to express the information that Radha has 15 notebooks. We may
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express it as [15] with the understanding that the number inside [ ] is the number of
notebooks that Radha has. Now, if we have to express that Radha has 15 notebooks
and 6 pens. We may express it as [15 6] with the understanding that first number
inside [ ] is the number of notebooks while the other one is the number of pens possessed
by Radha. Let us now suppose that we wish to express the information of possession
, MATRICES 57
of notebooks and pens by Radha and her two friends Fauzia and Simran which
is as follows:
Radha has 15 notebooks and 6 pens,
Fauzia has 10 notebooks and 2 pens,
d
Simran has 13 notebooks and 5 pens.
he
Now this could be arranged in the tabular form as follows:
Notebooks Pens
Radha 15 6
Fauzia 10 2
is
Simran 13 5
and this can be expressed as
bl
pu
be T
re
o R
or
Radha Fauzia Simran
tt E
Notebooks 15 10 13
Pens 6 2 5
C
which can be expressed as:
no N
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In the first arrangement the entries in the first column represent the number of
note books possessed by Radha, Fauzia and Simran, respectively and the entries in the
second column represent the number of pens possessed by Radha, Fauzia and Simran,
, 58 MATHEMATICS
respectively. Similarly, in the second arrangement, the entries in the first row represent
the number of notebooks possessed by Radha, Fauzia and Simran, respectively. The
entries in the second row represent the number of pens possessed by Radha, Fauzia
and Simran, respectively. An arrangement or display of the above kind is called a
d
matrix. Formally, we define matrix as:
Definition 1 A matrix is an ordered rectangular array of numbers or functions. The
he
numbers or functions are called the elements or the entries of the matrix.
We denote matrices by capital letters. The following are some examples of matrices:
⎡ 1⎤
⎡– 2 5⎤ ⎢2 + i 3 − 2 ⎥
is
⎢ ⎥ ⎢ ⎥ ⎡1 + x x3 3 ⎤
A=⎢ 0 5 ⎥ , B = ⎢ 3.5 –1 2 ⎥ , C = ⎢ ⎥
⎢ ⎥ ⎣ cos x sin x + 2 tan x ⎦
⎢3 ⎥ 5
⎣ 6⎦
bl
⎢ 3 5 ⎥
⎣ 7 ⎦
In the above examples, the horizontal lines of elements are said to constitute, rows
pu
of the matrix and the vertical lines of elements are said to constitute, columns of the
matrix. Thus A has 3 rows and 2 columns, B has 3 rows and 3 columns while C has 2
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rows and 3 columns.
re
3.2.1 Order of a matrix
o R
A matrix having m rows and n columns is called a matrix of order m × n or simply m × n
matrix (read as an m by n matrix). So referring to the above examples of matrices, we
have A as 3 × 2 matrix, B as 3 × 3 matrix and C as 2 × 3 matrix. We observe that A has
tt E
3 × 2 = 6 elements, B and C have 9 and 6 elements, respectively.
In general, an m × n matrix has the following rectangular array:
C
no N
©
or A = [aij]m × n, 1≤ i ≤ m, 1≤ j ≤ n i, j ∈ N
Thus the ith row consists of the elements ai1, ai2, ai3,..., ain, while the jth column
consists of the elements a1j, a2j, a3j,..., amj ,
In general aij, is an element lying in the ith row and jth column. We can also call
it as the (i, j)th element of A. The number of elements in an m × n matrix will be
equal to mn.
, MATRICES 59
$Note In this chapter
1. We shall follow the notation, namely A = [aij]m × n to indicate that A is a matrix
of order m × n.
d
2. We shall consider only those matrices whose elements are real numbers or
functions taking real values.
he
We can also represent any point (x, y) in a plane by a matrix (column or row) as
⎡x⎤
⎢ y ⎥ (or [x, y]). For example point P(0, 1) as a matrix representation may be given as
⎣ ⎦
is
⎡0 ⎤
P = ⎢ ⎥ or [0 1].
bl
⎣1 ⎦
Observe that in this way we can also express the vertices of a closed rectilinear
pu figure in the form of a matrix. For example, consider a quadrilateral ABCD with vertices
A (1, 0), B (3, 2), C (1, 3), D (–1, 2).
Now, quadrilateral ABCD in the matrix form, can be represented as
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A⎡ 1 0⎤
re
A B C D
B ⎢⎢ 3 2 ⎥⎥
o R
⎡1 3 1 −1⎤
X=⎢ ⎥ or Y=
⎣ 0 2 3 2⎦ 2 × 4 C⎢ 1 3⎥
⎢ ⎥
D ⎣−1
tt E
2 ⎦ 4× 2
Thus, matrices can be used as representation of vertices of geometrical figures in
C
a plane.
Now, let us consider some examples.
no N
Example 1 Consider the following information regarding the number of men and women
workers in three factories I, II and III
Men workers Women workers
©
I 30 25
II 25 31
III 27 26
Represent the above information in the form of a 3 × 2 matrix. What does the entry
in the third row and second column represent?
Chapter 3
MATRICES
d
he
The essence of Mathematics lies in its freedom. — CANTOR
3.1 Introduction
is
The knowledge of matrices is necessary in various branches of mathematics. Matrices
are one of the most powerful tools in mathematics. This mathematical tool simplifies
bl
our work to a great extent when compared with other straight forward methods. The
evolution of concept of matrices is the result of an attempt to obtain compact and
simple methods of solving system of linear equations. Matrices are not only used as a
pu
representation of the coefficients in system of linear equations, but utility of matrices
far exceeds that use. Matrix notation and operations are used in electronic spreadsheet
be T
programs for personal computer, which in turn is used in different areas of business
re
and science like budgeting, sales projection, cost estimation, analysing the results of an
o R
experiment etc. Also, many physical operations such as magnification, rotation and
reflection through a plane can be represented mathematically by matrices. Matrices
tt E
are also used in cryptography. This mathematical tool is not only used in certain branches
of sciences, but also in genetics, economics, sociology, modern psychology and industrial
C
management.
In this chapter, we shall find it interesting to become acquainted with the
no N
fundamentals of matrix and matrix algebra.
3.2 Matrix
Suppose we wish to express the information that Radha has 15 notebooks. We may
©
express it as [15] with the understanding that the number inside [ ] is the number of
notebooks that Radha has. Now, if we have to express that Radha has 15 notebooks
and 6 pens. We may express it as [15 6] with the understanding that first number
inside [ ] is the number of notebooks while the other one is the number of pens possessed
by Radha. Let us now suppose that we wish to express the information of possession
, MATRICES 57
of notebooks and pens by Radha and her two friends Fauzia and Simran which
is as follows:
Radha has 15 notebooks and 6 pens,
Fauzia has 10 notebooks and 2 pens,
d
Simran has 13 notebooks and 5 pens.
he
Now this could be arranged in the tabular form as follows:
Notebooks Pens
Radha 15 6
Fauzia 10 2
is
Simran 13 5
and this can be expressed as
bl
pu
be T
re
o R
or
Radha Fauzia Simran
tt E
Notebooks 15 10 13
Pens 6 2 5
C
which can be expressed as:
no N
©
In the first arrangement the entries in the first column represent the number of
note books possessed by Radha, Fauzia and Simran, respectively and the entries in the
second column represent the number of pens possessed by Radha, Fauzia and Simran,
, 58 MATHEMATICS
respectively. Similarly, in the second arrangement, the entries in the first row represent
the number of notebooks possessed by Radha, Fauzia and Simran, respectively. The
entries in the second row represent the number of pens possessed by Radha, Fauzia
and Simran, respectively. An arrangement or display of the above kind is called a
d
matrix. Formally, we define matrix as:
Definition 1 A matrix is an ordered rectangular array of numbers or functions. The
he
numbers or functions are called the elements or the entries of the matrix.
We denote matrices by capital letters. The following are some examples of matrices:
⎡ 1⎤
⎡– 2 5⎤ ⎢2 + i 3 − 2 ⎥
is
⎢ ⎥ ⎢ ⎥ ⎡1 + x x3 3 ⎤
A=⎢ 0 5 ⎥ , B = ⎢ 3.5 –1 2 ⎥ , C = ⎢ ⎥
⎢ ⎥ ⎣ cos x sin x + 2 tan x ⎦
⎢3 ⎥ 5
⎣ 6⎦
bl
⎢ 3 5 ⎥
⎣ 7 ⎦
In the above examples, the horizontal lines of elements are said to constitute, rows
pu
of the matrix and the vertical lines of elements are said to constitute, columns of the
matrix. Thus A has 3 rows and 2 columns, B has 3 rows and 3 columns while C has 2
be T
rows and 3 columns.
re
3.2.1 Order of a matrix
o R
A matrix having m rows and n columns is called a matrix of order m × n or simply m × n
matrix (read as an m by n matrix). So referring to the above examples of matrices, we
have A as 3 × 2 matrix, B as 3 × 3 matrix and C as 2 × 3 matrix. We observe that A has
tt E
3 × 2 = 6 elements, B and C have 9 and 6 elements, respectively.
In general, an m × n matrix has the following rectangular array:
C
no N
©
or A = [aij]m × n, 1≤ i ≤ m, 1≤ j ≤ n i, j ∈ N
Thus the ith row consists of the elements ai1, ai2, ai3,..., ain, while the jth column
consists of the elements a1j, a2j, a3j,..., amj ,
In general aij, is an element lying in the ith row and jth column. We can also call
it as the (i, j)th element of A. The number of elements in an m × n matrix will be
equal to mn.
, MATRICES 59
$Note In this chapter
1. We shall follow the notation, namely A = [aij]m × n to indicate that A is a matrix
of order m × n.
d
2. We shall consider only those matrices whose elements are real numbers or
functions taking real values.
he
We can also represent any point (x, y) in a plane by a matrix (column or row) as
⎡x⎤
⎢ y ⎥ (or [x, y]). For example point P(0, 1) as a matrix representation may be given as
⎣ ⎦
is
⎡0 ⎤
P = ⎢ ⎥ or [0 1].
bl
⎣1 ⎦
Observe that in this way we can also express the vertices of a closed rectilinear
pu figure in the form of a matrix. For example, consider a quadrilateral ABCD with vertices
A (1, 0), B (3, 2), C (1, 3), D (–1, 2).
Now, quadrilateral ABCD in the matrix form, can be represented as
be T
A⎡ 1 0⎤
re
A B C D
B ⎢⎢ 3 2 ⎥⎥
o R
⎡1 3 1 −1⎤
X=⎢ ⎥ or Y=
⎣ 0 2 3 2⎦ 2 × 4 C⎢ 1 3⎥
⎢ ⎥
D ⎣−1
tt E
2 ⎦ 4× 2
Thus, matrices can be used as representation of vertices of geometrical figures in
C
a plane.
Now, let us consider some examples.
no N
Example 1 Consider the following information regarding the number of men and women
workers in three factories I, II and III
Men workers Women workers
©
I 30 25
II 25 31
III 27 26
Represent the above information in the form of a 3 × 2 matrix. What does the entry
in the third row and second column represent?
