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1 | 1018-1021 | An arrangement or display of the above kind is called a
matrix Formally, we define matrix as:
Definition 1 A matrix is an ordered rectangular array of numbers or functions The
numbers or functions are called the elements or the entries of the matrix We denote matrices by capital letters |
1 | 1019-1022 | Formally, we define matrix as:
Definition 1 A matrix is an ordered rectangular array of numbers or functions The
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:
5
– 2
A
0
5
3
6
=
,
1
2
3
2
B
3 |
1 | 1020-1023 | The
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:
5
– 2
A
0
5
3
6
=
,
1
2
3
2
B
3 5
–1
2
5
3
5
7
i
+
−
=
,
3
1
3
C
cos
tan
sin
2
x
x
x
x
x
= +
+
In the above examples, the horizontal lines of elements are said to constitute, rows
of the matrix and the vertical lines of elements are said to constitute, columns of the
matrix |
1 | 1021-1024 | We denote matrices by capital letters The following are some examples of matrices:
5
– 2
A
0
5
3
6
=
,
1
2
3
2
B
3 5
–1
2
5
3
5
7
i
+
−
=
,
3
1
3
C
cos
tan
sin
2
x
x
x
x
x
= +
+
In the above examples, the horizontal lines of elements are said to constitute, rows
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
rows and 3 columns |
1 | 1022-1025 | The following are some examples of matrices:
5
– 2
A
0
5
3
6
=
,
1
2
3
2
B
3 5
–1
2
5
3
5
7
i
+
−
=
,
3
1
3
C
cos
tan
sin
2
x
x
x
x
x
= +
+
In the above examples, the horizontal lines of elements are said to constitute, rows
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
rows and 3 columns 3 |
1 | 1023-1026 | 5
–1
2
5
3
5
7
i
+
−
=
,
3
1
3
C
cos
tan
sin
2
x
x
x
x
x
= +
+
In the above examples, the horizontal lines of elements are said to constitute, rows
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
rows and 3 columns 3 2 |
1 | 1024-1027 | Thus A has 3 rows and 2 columns, B has 3 rows and 3 columns while C has 2
rows and 3 columns 3 2 1 Order of a matrix
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) |
1 | 1025-1028 | 3 2 1 Order of a matrix
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 |
1 | 1026-1029 | 2 1 Order of a matrix
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
3 × 2 = 6 elements, B and C have 9 and 6 elements, respectively |
1 | 1027-1030 | 1 Order of a matrix
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
3 × 2 = 6 elements, B and C have 9 and 6 elements, respectively In general, an m × n matrix has the following rectangular array:
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, |
1 | 1028-1031 | 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
3 × 2 = 6 elements, B and C have 9 and 6 elements, respectively In general, an m × n matrix has the following rectangular array:
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, |
1 | 1029-1032 | We observe that A has
3 × 2 = 6 elements, B and C have 9 and 6 elements, respectively In general, an m × n matrix has the following rectangular array:
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 |
1 | 1030-1033 | In general, an m × n matrix has the following rectangular array:
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 |
1 | 1031-1034 | , 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 |
1 | 1032-1035 | , 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 Rationalised 2023-24
MATRICES 37
ANote In this chapter
1 |
1 | 1033-1036 | 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 Rationalised 2023-24
MATRICES 37
ANote 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 |
1 | 1034-1037 | The number of elements in an m × n matrix will be
equal to mn Rationalised 2023-24
MATRICES 37
ANote 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 2 |
1 | 1035-1038 | Rationalised 2023-24
MATRICES 37
ANote 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 2 We shall consider only those matrices whose elements are real numbers or
functions taking real values |
1 | 1036-1039 | We shall follow the notation, namely A = [aij]m × n to indicate that A is a matrix
of order m × n 2 We shall consider only those matrices whose elements are real numbers or
functions taking real values We can also represent any point (x, y) in a plane by a matrix (column or row) as
x
y
(or [x, y]) |
1 | 1037-1040 | 2 We shall consider only those matrices whose elements are real numbers or
functions taking real values 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
0
P
1
=
or [0 1] |
1 | 1038-1041 | We shall consider only those matrices whose elements are real numbers or
functions taking real values 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
0
P
1
=
or [0 1] Observe that in this way we can also express the vertices of a closed rectilinear
figure in the form of a matrix |
1 | 1039-1042 | 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
0
P
1
=
or [0 1] Observe that in this way we can also express the vertices of a closed rectilinear
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) |
1 | 1040-1043 | For example point P(0, 1) as a matrix representation may be given as
0
P
1
=
or [0 1] Observe that in this way we can also express the vertices of a closed rectilinear
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
2
4
A
B
C D
1
3
1
1
X
0 2
3
2
×
−
=
or
4 2
A 1
0
B
3
2
Y
C
1
3
D
1
2
×
=
−
Thus, matrices can be used as representation of vertices of geometrical figures in
a plane |
1 | 1041-1044 | Observe that in this way we can also express the vertices of a closed rectilinear
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
2
4
A
B
C D
1
3
1
1
X
0 2
3
2
×
−
=
or
4 2
A 1
0
B
3
2
Y
C
1
3
D
1
2
×
=
−
Thus, matrices can be used as representation of vertices of geometrical figures in
a plane Now, let us consider some examples |
1 | 1042-1045 | 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
2
4
A
B
C D
1
3
1
1
X
0 2
3
2
×
−
=
or
4 2
A 1
0
B
3
2
Y
C
1
3
D
1
2
×
=
−
Thus, matrices can be used as representation of vertices of geometrical figures in
a plane Now, let us consider some examples 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 |
1 | 1043-1046 | Now, quadrilateral ABCD in the matrix form, can be represented as
2
4
A
B
C D
1
3
1
1
X
0 2
3
2
×
−
=
or
4 2
A 1
0
B
3
2
Y
C
1
3
D
1
2
×
=
−
Thus, matrices can be used as representation of vertices of geometrical figures in
a plane Now, let us consider some examples 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 |
1 | 1044-1047 | Now, let us consider some examples 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 Rationalised 2023-24
38
MATHEMATICS
Solution The information is represented in the form of a 3 × 2 matrix as follows:
30
25
A
25
31
27
26
=
The entry in the third row and second column represents the number of women
workers in factory III |
1 | 1045-1048 | 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 Rationalised 2023-24
38
MATHEMATICS
Solution The information is represented in the form of a 3 × 2 matrix as follows:
30
25
A
25
31
27
26
=
The entry in the third row and second column represents the number of women
workers in factory III Example 2 If a matrix has 8 elements, what are the possible orders it can have |
1 | 1046-1049 | What does the entry
in the third row and second column represent Rationalised 2023-24
38
MATHEMATICS
Solution The information is represented in the form of a 3 × 2 matrix as follows:
30
25
A
25
31
27
26
=
The entry in the third row and second column represents the number of women
workers in factory III Example 2 If a matrix has 8 elements, what are the possible orders it can have Solution We know that if a matrix is of order m × n, it has mn elements |
1 | 1047-1050 | Rationalised 2023-24
38
MATHEMATICS
Solution The information is represented in the form of a 3 × 2 matrix as follows:
30
25
A
25
31
27
26
=
The entry in the third row and second column represents the number of women
workers in factory III Example 2 If a matrix has 8 elements, what are the possible orders it can have Solution We know that if a matrix is of order m × n, it has mn elements Thus, to find
all possible orders of a matrix with 8 elements, we will find all ordered pairs of natural
numbers, whose product is 8 |
1 | 1048-1051 | Example 2 If a matrix has 8 elements, what are the possible orders it can have Solution We know that if a matrix is of order m × n, it has mn elements Thus, to find
all possible orders of a matrix with 8 elements, we will find all ordered pairs of natural
numbers, whose product is 8 Thus, all possible ordered pairs are (1, 8), (8, 1), (4, 2), (2, 4)
Hence, possible orders are 1 × 8, 8 ×1, 4 × 2, 2 × 4
Example 3 Construct a 3 × 2 matrix whose elements are given by
1 |
3 |
2
aij
i
j
=
− |
1 | 1049-1052 | Solution We know that if a matrix is of order m × n, it has mn elements Thus, to find
all possible orders of a matrix with 8 elements, we will find all ordered pairs of natural
numbers, whose product is 8 Thus, all possible ordered pairs are (1, 8), (8, 1), (4, 2), (2, 4)
Hence, possible orders are 1 × 8, 8 ×1, 4 × 2, 2 × 4
Example 3 Construct a 3 × 2 matrix whose elements are given by
1 |
3 |
2
aij
i
j
=
− Solution In general a 3 × 2 matrix is given by
11
12
21
22
31
32
A
a
a
a
a
a
a
=
|
1 | 1050-1053 | Thus, to find
all possible orders of a matrix with 8 elements, we will find all ordered pairs of natural
numbers, whose product is 8 Thus, all possible ordered pairs are (1, 8), (8, 1), (4, 2), (2, 4)
Hence, possible orders are 1 × 8, 8 ×1, 4 × 2, 2 × 4
Example 3 Construct a 3 × 2 matrix whose elements are given by
1 |
3 |
2
aij
i
j
=
− Solution In general a 3 × 2 matrix is given by
11
12
21
22
31
32
A
a
a
a
a
a
a
=
Now
1 |
3 |
2
aij
i
j
=
−
, i = 1, 2, 3 and j = 1, 2 |
1 | 1051-1054 | Thus, all possible ordered pairs are (1, 8), (8, 1), (4, 2), (2, 4)
Hence, possible orders are 1 × 8, 8 ×1, 4 × 2, 2 × 4
Example 3 Construct a 3 × 2 matrix whose elements are given by
1 |
3 |
2
aij
i
j
=
− Solution In general a 3 × 2 matrix is given by
11
12
21
22
31
32
A
a
a
a
a
a
a
=
Now
1 |
3 |
2
aij
i
j
=
−
, i = 1, 2, 3 and j = 1, 2 Therefore
11
1 |1 3 1|
1
2
a
=
− ×
=
12
1
5
2|1 3 2|
2
a
=
− ×
=
21
1
1
| 2
3 1|
2
2
a
=
− ×
=
22
1 | 2
3 2|
2
2
a
=
− ×
=
31
1 |3
3 1|
0
2
a
=
− ×
=
32
1
3
|3
3 2 |
2
2
a
=
− ×
=
Hence the required matrix is given by
5
1
2
1
A
2
2
3
0
2
=
|
1 | 1052-1055 | Solution In general a 3 × 2 matrix is given by
11
12
21
22
31
32
A
a
a
a
a
a
a
=
Now
1 |
3 |
2
aij
i
j
=
−
, i = 1, 2, 3 and j = 1, 2 Therefore
11
1 |1 3 1|
1
2
a
=
− ×
=
12
1
5
2|1 3 2|
2
a
=
− ×
=
21
1
1
| 2
3 1|
2
2
a
=
− ×
=
22
1 | 2
3 2|
2
2
a
=
− ×
=
31
1 |3
3 1|
0
2
a
=
− ×
=
32
1
3
|3
3 2 |
2
2
a
=
− ×
=
Hence the required matrix is given by
5
1
2
1
A
2
2
3
0
2
=
Rationalised 2023-24
MATRICES 39
3 |
1 | 1053-1056 | Now
1 |
3 |
2
aij
i
j
=
−
, i = 1, 2, 3 and j = 1, 2 Therefore
11
1 |1 3 1|
1
2
a
=
− ×
=
12
1
5
2|1 3 2|
2
a
=
− ×
=
21
1
1
| 2
3 1|
2
2
a
=
− ×
=
22
1 | 2
3 2|
2
2
a
=
− ×
=
31
1 |3
3 1|
0
2
a
=
− ×
=
32
1
3
|3
3 2 |
2
2
a
=
− ×
=
Hence the required matrix is given by
5
1
2
1
A
2
2
3
0
2
=
Rationalised 2023-24
MATRICES 39
3 3 Types of Matrices
In this section, we shall discuss different types of matrices |
1 | 1054-1057 | Therefore
11
1 |1 3 1|
1
2
a
=
− ×
=
12
1
5
2|1 3 2|
2
a
=
− ×
=
21
1
1
| 2
3 1|
2
2
a
=
− ×
=
22
1 | 2
3 2|
2
2
a
=
− ×
=
31
1 |3
3 1|
0
2
a
=
− ×
=
32
1
3
|3
3 2 |
2
2
a
=
− ×
=
Hence the required matrix is given by
5
1
2
1
A
2
2
3
0
2
=
Rationalised 2023-24
MATRICES 39
3 3 Types of Matrices
In this section, we shall discuss different types of matrices (i)
Column matrix
A matrix is said to be a column matrix if it has only one column |
1 | 1055-1058 | Rationalised 2023-24
MATRICES 39
3 3 Types of Matrices
In this section, we shall discuss different types of matrices (i)
Column matrix
A matrix is said to be a column matrix if it has only one column For example,
0
3
A
1
1/ 2
=
−
is a column matrix of order 4 × 1 |
1 | 1056-1059 | 3 Types of Matrices
In this section, we shall discuss different types of matrices (i)
Column matrix
A matrix is said to be a column matrix if it has only one column For example,
0
3
A
1
1/ 2
=
−
is a column matrix of order 4 × 1 In general, A = [aij] m × 1 is a column matrix of order m × 1 |
1 | 1057-1060 | (i)
Column matrix
A matrix is said to be a column matrix if it has only one column For example,
0
3
A
1
1/ 2
=
−
is a column matrix of order 4 × 1 In general, A = [aij] m × 1 is a column matrix of order m × 1 (ii)
Row matrix
A matrix is said to be a row matrix if it has only one row |
1 | 1058-1061 | For example,
0
3
A
1
1/ 2
=
−
is a column matrix of order 4 × 1 In general, A = [aij] m × 1 is a column matrix of order m × 1 (ii)
Row matrix
A matrix is said to be a row matrix if it has only one row For example,
1 4
1
B
5 2 3
2
×
= −
is a row matrix |
1 | 1059-1062 | In general, A = [aij] m × 1 is a column matrix of order m × 1 (ii)
Row matrix
A matrix is said to be a row matrix if it has only one row For example,
1 4
1
B
5 2 3
2
×
= −
is a row matrix In general, B = [bij] 1 × n is a row matrix of order 1 × n |
1 | 1060-1063 | (ii)
Row matrix
A matrix is said to be a row matrix if it has only one row For example,
1 4
1
B
5 2 3
2
×
= −
is a row matrix In general, B = [bij] 1 × n is a row matrix of order 1 × n (iii)
Square matrix
A matrix in which the number of rows are equal to the number of columns, is
said to be a square matrix |
1 | 1061-1064 | For example,
1 4
1
B
5 2 3
2
×
= −
is a row matrix In general, B = [bij] 1 × n is a row matrix of order 1 × n (iii)
Square matrix
A matrix in which the number of rows are equal to the number of columns, is
said to be a square matrix Thus an m × n matrix is said to be a square matrix if
m = n and is known as a square matrix of order ‘n’ |
1 | 1062-1065 | In general, B = [bij] 1 × n is a row matrix of order 1 × n (iii)
Square matrix
A matrix in which the number of rows are equal to the number of columns, is
said to be a square matrix Thus an m × n matrix is said to be a square matrix if
m = n and is known as a square matrix of order ‘n’ For example
3
1
0
3
A
3 2
1
42
3
1
−
=
−
is a square matrix of order 3 |
1 | 1063-1066 | (iii)
Square matrix
A matrix in which the number of rows are equal to the number of columns, is
said to be a square matrix Thus an m × n matrix is said to be a square matrix if
m = n and is known as a square matrix of order ‘n’ For example
3
1
0
3
A
3 2
1
42
3
1
−
=
−
is a square matrix of order 3 In general, A = [aij] m × m is a square matrix of order m |
1 | 1064-1067 | Thus an m × n matrix is said to be a square matrix if
m = n and is known as a square matrix of order ‘n’ For example
3
1
0
3
A
3 2
1
42
3
1
−
=
−
is a square matrix of order 3 In general, A = [aij] m × m is a square matrix of order m ANote If A = [aij] is a square matrix of order n, then elements (entries) a11, a22, |
1 | 1065-1068 | For example
3
1
0
3
A
3 2
1
42
3
1
−
=
−
is a square matrix of order 3 In general, A = [aij] m × m is a square matrix of order m ANote If A = [aij] is a square matrix of order n, then elements (entries) a11, a22, , ann
are said to constitute the diagonal, of the matrix A |
1 | 1066-1069 | In general, A = [aij] m × m is a square matrix of order m ANote If A = [aij] is a square matrix of order n, then elements (entries) a11, a22, , ann
are said to constitute the diagonal, of the matrix A Thus, if
1
3
1
A
2
4
1
3
5
6
−
=
−
|
1 | 1067-1070 | ANote If A = [aij] is a square matrix of order n, then elements (entries) a11, a22, , ann
are said to constitute the diagonal, of the matrix A Thus, if
1
3
1
A
2
4
1
3
5
6
−
=
−
Then the elements of the diagonal of A are 1, 4, 6 |
1 | 1068-1071 | , ann
are said to constitute the diagonal, of the matrix A Thus, if
1
3
1
A
2
4
1
3
5
6
−
=
−
Then the elements of the diagonal of A are 1, 4, 6 Rationalised 2023-24
40
MATHEMATICS
(iv)
Diagonal matrix
A square matrix B = [bij] m × m is said to be a diagonal matrix if all its non
diagonal elements are zero, that is a matrix B = [bij] m × m is said to be a diagonal
matrix if bij = 0, when i ≠ j |
1 | 1069-1072 | Thus, if
1
3
1
A
2
4
1
3
5
6
−
=
−
Then the elements of the diagonal of A are 1, 4, 6 Rationalised 2023-24
40
MATHEMATICS
(iv)
Diagonal matrix
A square matrix B = [bij] m × m is said to be a diagonal matrix if all its non
diagonal elements are zero, that is a matrix B = [bij] m × m is said to be a diagonal
matrix if bij = 0, when i ≠ j For example, A = [4],
1
0
B
0
2
−
=
,
1 |
1 | 1070-1073 | Then the elements of the diagonal of A are 1, 4, 6 Rationalised 2023-24
40
MATHEMATICS
(iv)
Diagonal matrix
A square matrix B = [bij] m × m is said to be a diagonal matrix if all its non
diagonal elements are zero, that is a matrix B = [bij] m × m is said to be a diagonal
matrix if bij = 0, when i ≠ j For example, A = [4],
1
0
B
0
2
−
=
,
1 1
0
0
C
0
2
0
0
0
3
−
=
, are diagonal matrices
of order 1, 2, 3, respectively |
1 | 1071-1074 | Rationalised 2023-24
40
MATHEMATICS
(iv)
Diagonal matrix
A square matrix B = [bij] m × m is said to be a diagonal matrix if all its non
diagonal elements are zero, that is a matrix B = [bij] m × m is said to be a diagonal
matrix if bij = 0, when i ≠ j For example, A = [4],
1
0
B
0
2
−
=
,
1 1
0
0
C
0
2
0
0
0
3
−
=
, are diagonal matrices
of order 1, 2, 3, respectively (v)
Scalar matrix
A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal,
that is, a square matrix B = [bij] n × n is said to be a scalar matrix if
bij = 0, when i ≠ j
bij = k, when i = j, for some constant k |
1 | 1072-1075 | For example, A = [4],
1
0
B
0
2
−
=
,
1 1
0
0
C
0
2
0
0
0
3
−
=
, are diagonal matrices
of order 1, 2, 3, respectively (v)
Scalar matrix
A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal,
that is, a square matrix B = [bij] n × n is said to be a scalar matrix if
bij = 0, when i ≠ j
bij = k, when i = j, for some constant k For example
A = [3],
1
0
B
0
1
−
=
−
,
3
0
0
C
0
3
0
0
0
3
=
are scalar matrices of order 1, 2 and 3, respectively |
1 | 1073-1076 | 1
0
0
C
0
2
0
0
0
3
−
=
, are diagonal matrices
of order 1, 2, 3, respectively (v)
Scalar matrix
A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal,
that is, a square matrix B = [bij] n × n is said to be a scalar matrix if
bij = 0, when i ≠ j
bij = k, when i = j, for some constant k For example
A = [3],
1
0
B
0
1
−
=
−
,
3
0
0
C
0
3
0
0
0
3
=
are scalar matrices of order 1, 2 and 3, respectively (vi)
Identity matrix
A square matrix in which elements in the diagonal are all 1 and rest are all zero
is called an identity matrix |
1 | 1074-1077 | (v)
Scalar matrix
A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal,
that is, a square matrix B = [bij] n × n is said to be a scalar matrix if
bij = 0, when i ≠ j
bij = k, when i = j, for some constant k For example
A = [3],
1
0
B
0
1
−
=
−
,
3
0
0
C
0
3
0
0
0
3
=
are scalar matrices of order 1, 2 and 3, respectively (vi)
Identity matrix
A square matrix in which elements in the diagonal are all 1 and rest are all zero
is called an identity matrix In other words, the square matrix A = [aij] n × n is an
identity matrix, if
1
0if
if
ij
i
j
a
i
j
=
=
≠
|
1 | 1075-1078 | For example
A = [3],
1
0
B
0
1
−
=
−
,
3
0
0
C
0
3
0
0
0
3
=
are scalar matrices of order 1, 2 and 3, respectively (vi)
Identity matrix
A square matrix in which elements in the diagonal are all 1 and rest are all zero
is called an identity matrix In other words, the square matrix A = [aij] n × n is an
identity matrix, if
1
0if
if
ij
i
j
a
i
j
=
=
≠
We denote the identity matrix of order n by In |
1 | 1076-1079 | (vi)
Identity matrix
A square matrix in which elements in the diagonal are all 1 and rest are all zero
is called an identity matrix In other words, the square matrix A = [aij] n × n is an
identity matrix, if
1
0if
if
ij
i
j
a
i
j
=
=
≠
We denote the identity matrix of order n by In When order is clear from the
context, we simply write it as I |
1 | 1077-1080 | In other words, the square matrix A = [aij] n × n is an
identity matrix, if
1
0if
if
ij
i
j
a
i
j
=
=
≠
We denote the identity matrix of order n by In When order is clear from the
context, we simply write it as I For example [1],
1
0
0
1
,
1
0
0
0
1
0
0
0
1
are identity matrices of order 1, 2 and 3,
respectively |
1 | 1078-1081 | We denote the identity matrix of order n by In When order is clear from the
context, we simply write it as I For example [1],
1
0
0
1
,
1
0
0
0
1
0
0
0
1
are identity matrices of order 1, 2 and 3,
respectively Observe that a scalar matrix is an identity matrix when k = 1 |
1 | 1079-1082 | When order is clear from the
context, we simply write it as I For example [1],
1
0
0
1
,
1
0
0
0
1
0
0
0
1
are identity matrices of order 1, 2 and 3,
respectively Observe that a scalar matrix is an identity matrix when k = 1 But every identity
matrix is clearly a scalar matrix |
1 | 1080-1083 | For example [1],
1
0
0
1
,
1
0
0
0
1
0
0
0
1
are identity matrices of order 1, 2 and 3,
respectively Observe that a scalar matrix is an identity matrix when k = 1 But every identity
matrix is clearly a scalar matrix Rationalised 2023-24
MATRICES 41
(vii)
Zero matrix
A matrix is said to be zero matrix or null matrix if all its elements are zero |
1 | 1081-1084 | Observe that a scalar matrix is an identity matrix when k = 1 But every identity
matrix is clearly a scalar matrix Rationalised 2023-24
MATRICES 41
(vii)
Zero matrix
A matrix is said to be zero matrix or null matrix if all its elements are zero For example, [0],
0
0
0
0
,
0
0
0
0
0
0
, [0, 0] are all zero matrices |
1 | 1082-1085 | But every identity
matrix is clearly a scalar matrix Rationalised 2023-24
MATRICES 41
(vii)
Zero matrix
A matrix is said to be zero matrix or null matrix if all its elements are zero For example, [0],
0
0
0
0
,
0
0
0
0
0
0
, [0, 0] are all zero matrices We denote
zero matrix by O |
1 | 1083-1086 | Rationalised 2023-24
MATRICES 41
(vii)
Zero matrix
A matrix is said to be zero matrix or null matrix if all its elements are zero For example, [0],
0
0
0
0
,
0
0
0
0
0
0
, [0, 0] are all zero matrices We denote
zero matrix by O Its order will be clear from the context |
1 | 1084-1087 | For example, [0],
0
0
0
0
,
0
0
0
0
0
0
, [0, 0] are all zero matrices We denote
zero matrix by O Its order will be clear from the context 3 |
1 | 1085-1088 | We denote
zero matrix by O Its order will be clear from the context 3 3 |
1 | 1086-1089 | Its order will be clear from the context 3 3 1 Equality of matrices
Definition 2 Two matrices A = [aij] and B = [bij] are said to be equal if
(i)
they are of the same order
(ii)
each element of A is equal to the corresponding element of B, that is aij = bij for
all i and j |
1 | 1087-1090 | 3 3 1 Equality of matrices
Definition 2 Two matrices A = [aij] and B = [bij] are said to be equal if
(i)
they are of the same order
(ii)
each element of A is equal to the corresponding element of B, that is aij = bij for
all i and j For example, 2
3
2
3
and
0
1
0
1
are equal matrices but 3
2
2
3
and
0
1
0
1
are
not equal matrices |
1 | 1088-1091 | 3 1 Equality of matrices
Definition 2 Two matrices A = [aij] and B = [bij] are said to be equal if
(i)
they are of the same order
(ii)
each element of A is equal to the corresponding element of B, that is aij = bij for
all i and j For example, 2
3
2
3
and
0
1
0
1
are equal matrices but 3
2
2
3
and
0
1
0
1
are
not equal matrices Symbolically, if two matrices A and B are equal, we write A = B |
1 | 1089-1092 | 1 Equality of matrices
Definition 2 Two matrices A = [aij] and B = [bij] are said to be equal if
(i)
they are of the same order
(ii)
each element of A is equal to the corresponding element of B, that is aij = bij for
all i and j For example, 2
3
2
3
and
0
1
0
1
are equal matrices but 3
2
2
3
and
0
1
0
1
are
not equal matrices Symbolically, if two matrices A and B are equal, we write A = B If
1 |
1 | 1090-1093 | For example, 2
3
2
3
and
0
1
0
1
are equal matrices but 3
2
2
3
and
0
1
0
1
are
not equal matrices Symbolically, if two matrices A and B are equal, we write A = B If
1 5
0
2
6
3
2
x
y
z
a
b
c
−
=
, then x = – 1 |
1 | 1091-1094 | Symbolically, if two matrices A and B are equal, we write A = B If
1 5
0
2
6
3
2
x
y
z
a
b
c
−
=
, then x = – 1 5, y = 0, z = 2, a =
6 , b = 3, c = 2
Example 4 If
3
4
2
7
0
6
3
2
6
1
0
6
3
2
2
3
21
0
2
4
21
0
x
z
y
y
a
c
b
b
+
+
−
−
−
−
= −
−
+
−
−
+
−
Find the values of a, b, c, x, y and z |
1 | 1092-1095 | If
1 5
0
2
6
3
2
x
y
z
a
b
c
−
=
, then x = – 1 5, y = 0, z = 2, a =
6 , b = 3, c = 2
Example 4 If
3
4
2
7
0
6
3
2
6
1
0
6
3
2
2
3
21
0
2
4
21
0
x
z
y
y
a
c
b
b
+
+
−
−
−
−
= −
−
+
−
−
+
−
Find the values of a, b, c, x, y and z Solution As the given matrices are equal, therefore, their corresponding elements
must be equal |
1 | 1093-1096 | 5
0
2
6
3
2
x
y
z
a
b
c
−
=
, then x = – 1 5, y = 0, z = 2, a =
6 , b = 3, c = 2
Example 4 If
3
4
2
7
0
6
3
2
6
1
0
6
3
2
2
3
21
0
2
4
21
0
x
z
y
y
a
c
b
b
+
+
−
−
−
−
= −
−
+
−
−
+
−
Find the values of a, b, c, x, y and z Solution As the given matrices are equal, therefore, their corresponding elements
must be equal Comparing the corresponding elements, we get
x + 3 = 0,
z + 4 = 6,
2y – 7 = 3y – 2
a – 1 = – 3,
0 = 2c + 2
b – 3 = 2b + 4,
Simplifying, we get
a = – 2, b = – 7, c = – 1, x = – 3, y = –5, z = 2
Example 5 Find the values of a, b, c, and d from the following equation:
2
2
4
3
5
4
3
11
24
a
b
a
b
c
d
c
d
+
−
−
=
−
+
Rationalised 2023-24
42
MATHEMATICS
Solution By equality of two matrices, equating the corresponding elements, we get
2a + b = 4
5c – d = 11
a – 2b = – 3
4c + 3d = 24
Solving these equations, we get
a = 1, b = 2, c = 3 and d = 4
EXERCISE 3 |
1 | 1094-1097 | 5, y = 0, z = 2, a =
6 , b = 3, c = 2
Example 4 If
3
4
2
7
0
6
3
2
6
1
0
6
3
2
2
3
21
0
2
4
21
0
x
z
y
y
a
c
b
b
+
+
−
−
−
−
= −
−
+
−
−
+
−
Find the values of a, b, c, x, y and z Solution As the given matrices are equal, therefore, their corresponding elements
must be equal Comparing the corresponding elements, we get
x + 3 = 0,
z + 4 = 6,
2y – 7 = 3y – 2
a – 1 = – 3,
0 = 2c + 2
b – 3 = 2b + 4,
Simplifying, we get
a = – 2, b = – 7, c = – 1, x = – 3, y = –5, z = 2
Example 5 Find the values of a, b, c, and d from the following equation:
2
2
4
3
5
4
3
11
24
a
b
a
b
c
d
c
d
+
−
−
=
−
+
Rationalised 2023-24
42
MATHEMATICS
Solution By equality of two matrices, equating the corresponding elements, we get
2a + b = 4
5c – d = 11
a – 2b = – 3
4c + 3d = 24
Solving these equations, we get
a = 1, b = 2, c = 3 and d = 4
EXERCISE 3 1
1 |
1 | 1095-1098 | Solution As the given matrices are equal, therefore, their corresponding elements
must be equal Comparing the corresponding elements, we get
x + 3 = 0,
z + 4 = 6,
2y – 7 = 3y – 2
a – 1 = – 3,
0 = 2c + 2
b – 3 = 2b + 4,
Simplifying, we get
a = – 2, b = – 7, c = – 1, x = – 3, y = –5, z = 2
Example 5 Find the values of a, b, c, and d from the following equation:
2
2
4
3
5
4
3
11
24
a
b
a
b
c
d
c
d
+
−
−
=
−
+
Rationalised 2023-24
42
MATHEMATICS
Solution By equality of two matrices, equating the corresponding elements, we get
2a + b = 4
5c – d = 11
a – 2b = – 3
4c + 3d = 24
Solving these equations, we get
a = 1, b = 2, c = 3 and d = 4
EXERCISE 3 1
1 In the matrix
2
5
19
7
5
A
35
2
12
2
17
3
1
5
−
=
−
−
, write:
(i) The order of the matrix,
(ii) The number of elements,
(iii) Write the elements a13, a21, a33, a24, a23 |
1 | 1096-1099 | Comparing the corresponding elements, we get
x + 3 = 0,
z + 4 = 6,
2y – 7 = 3y – 2
a – 1 = – 3,
0 = 2c + 2
b – 3 = 2b + 4,
Simplifying, we get
a = – 2, b = – 7, c = – 1, x = – 3, y = –5, z = 2
Example 5 Find the values of a, b, c, and d from the following equation:
2
2
4
3
5
4
3
11
24
a
b
a
b
c
d
c
d
+
−
−
=
−
+
Rationalised 2023-24
42
MATHEMATICS
Solution By equality of two matrices, equating the corresponding elements, we get
2a + b = 4
5c – d = 11
a – 2b = – 3
4c + 3d = 24
Solving these equations, we get
a = 1, b = 2, c = 3 and d = 4
EXERCISE 3 1
1 In the matrix
2
5
19
7
5
A
35
2
12
2
17
3
1
5
−
=
−
−
, write:
(i) The order of the matrix,
(ii) The number of elements,
(iii) Write the elements a13, a21, a33, a24, a23 2 |
1 | 1097-1100 | 1
1 In the matrix
2
5
19
7
5
A
35
2
12
2
17
3
1
5
−
=
−
−
, write:
(i) The order of the matrix,
(ii) The number of elements,
(iii) Write the elements a13, a21, a33, a24, a23 2 If a matrix has 24 elements, what are the possible orders it can have |
1 | 1098-1101 | In the matrix
2
5
19
7
5
A
35
2
12
2
17
3
1
5
−
=
−
−
, write:
(i) The order of the matrix,
(ii) The number of elements,
(iii) Write the elements a13, a21, a33, a24, a23 2 If a matrix has 24 elements, what are the possible orders it can have What, if it
has 13 elements |
1 | 1099-1102 | 2 If a matrix has 24 elements, what are the possible orders it can have What, if it
has 13 elements 3 |
1 | 1100-1103 | If a matrix has 24 elements, what are the possible orders it can have What, if it
has 13 elements 3 If a matrix has 18 elements, what are the possible orders it can have |
1 | 1101-1104 | What, if it
has 13 elements 3 If a matrix has 18 elements, what are the possible orders it can have What, if it
has 5 elements |
1 | 1102-1105 | 3 If a matrix has 18 elements, what are the possible orders it can have What, if it
has 5 elements 4 |
1 | 1103-1106 | If a matrix has 18 elements, what are the possible orders it can have What, if it
has 5 elements 4 Construct a 2 × 2 matrix, A = [aij], whose elements are given by:
(i)
2
(
)
2
ij
i
j
a
+
=
(ii)
ij
i
a
j
=
(iii)
2
(
2 )
2
ij
i
j
a
+
=
5 |
1 | 1104-1107 | What, if it
has 5 elements 4 Construct a 2 × 2 matrix, A = [aij], whose elements are given by:
(i)
2
(
)
2
ij
i
j
a
+
=
(ii)
ij
i
a
j
=
(iii)
2
(
2 )
2
ij
i
j
a
+
=
5 Construct a 3 × 4 matrix, whose elements are given by:
(i)
1 | 3
|
2
aij
i
j
=
−
+
(ii)
2
aij
i
j
=
−
6 |
1 | 1105-1108 | 4 Construct a 2 × 2 matrix, A = [aij], whose elements are given by:
(i)
2
(
)
2
ij
i
j
a
+
=
(ii)
ij
i
a
j
=
(iii)
2
(
2 )
2
ij
i
j
a
+
=
5 Construct a 3 × 4 matrix, whose elements are given by:
(i)
1 | 3
|
2
aij
i
j
=
−
+
(ii)
2
aij
i
j
=
−
6 Find the values of x, y and z from the following equations:
(i)
4
3
5
1
5
y
z
x
=
(ii)
2
6
2
5
5
8
x
y
z
xy
+
=
+
(iii)
9
5
7
x
y
z
x
z
y
z
+
+
+
=
+
7 |
1 | 1106-1109 | Construct a 2 × 2 matrix, A = [aij], whose elements are given by:
(i)
2
(
)
2
ij
i
j
a
+
=
(ii)
ij
i
a
j
=
(iii)
2
(
2 )
2
ij
i
j
a
+
=
5 Construct a 3 × 4 matrix, whose elements are given by:
(i)
1 | 3
|
2
aij
i
j
=
−
+
(ii)
2
aij
i
j
=
−
6 Find the values of x, y and z from the following equations:
(i)
4
3
5
1
5
y
z
x
=
(ii)
2
6
2
5
5
8
x
y
z
xy
+
=
+
(iii)
9
5
7
x
y
z
x
z
y
z
+
+
+
=
+
7 Find the value of a, b, c and d from the equation:
2
1
5
2
3
0
13
a
b
a
c
a
b
c
d
−
+
−
=
−
+
Rationalised 2023-24
MATRICES 43
8 |
1 | 1107-1110 | Construct a 3 × 4 matrix, whose elements are given by:
(i)
1 | 3
|
2
aij
i
j
=
−
+
(ii)
2
aij
i
j
=
−
6 Find the values of x, y and z from the following equations:
(i)
4
3
5
1
5
y
z
x
=
(ii)
2
6
2
5
5
8
x
y
z
xy
+
=
+
(iii)
9
5
7
x
y
z
x
z
y
z
+
+
+
=
+
7 Find the value of a, b, c and d from the equation:
2
1
5
2
3
0
13
a
b
a
c
a
b
c
d
−
+
−
=
−
+
Rationalised 2023-24
MATRICES 43
8 A = [aij]m × n\ is a square matrix, if
(A) m < n
(B) m > n
(C) m = n
(D) None of these
9 |
1 | 1108-1111 | Find the values of x, y and z from the following equations:
(i)
4
3
5
1
5
y
z
x
=
(ii)
2
6
2
5
5
8
x
y
z
xy
+
=
+
(iii)
9
5
7
x
y
z
x
z
y
z
+
+
+
=
+
7 Find the value of a, b, c and d from the equation:
2
1
5
2
3
0
13
a
b
a
c
a
b
c
d
−
+
−
=
−
+
Rationalised 2023-24
MATRICES 43
8 A = [aij]m × n\ is a square matrix, if
(A) m < n
(B) m > n
(C) m = n
(D) None of these
9 Which of the given values of x and y make the following pair of matrices equal
3
7
5
1
2
3
x
y
x
+
+
−
,
0
2
8
4
y −
(A)
1,
7
3
x
y
=−
=
(B) Not possible to find
(C) y = 7,
32
x
=−
(D)
1
2
3,
3
x
y
−
−
=
=
10 |
1 | 1109-1112 | Find the value of a, b, c and d from the equation:
2
1
5
2
3
0
13
a
b
a
c
a
b
c
d
−
+
−
=
−
+
Rationalised 2023-24
MATRICES 43
8 A = [aij]m × n\ is a square matrix, if
(A) m < n
(B) m > n
(C) m = n
(D) None of these
9 Which of the given values of x and y make the following pair of matrices equal
3
7
5
1
2
3
x
y
x
+
+
−
,
0
2
8
4
y −
(A)
1,
7
3
x
y
=−
=
(B) Not possible to find
(C) y = 7,
32
x
=−
(D)
1
2
3,
3
x
y
−
−
=
=
10 The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:
(A) 27
(B) 18
(C) 81
(D) 512
3 |
1 | 1110-1113 | A = [aij]m × n\ is a square matrix, if
(A) m < n
(B) m > n
(C) m = n
(D) None of these
9 Which of the given values of x and y make the following pair of matrices equal
3
7
5
1
2
3
x
y
x
+
+
−
,
0
2
8
4
y −
(A)
1,
7
3
x
y
=−
=
(B) Not possible to find
(C) y = 7,
32
x
=−
(D)
1
2
3,
3
x
y
−
−
=
=
10 The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:
(A) 27
(B) 18
(C) 81
(D) 512
3 4 Operations on Matrices
In this section, we shall introduce certain operations on matrices, namely, addition of
matrices, multiplication of a matrix by a scalar, difference and multiplication of matrices |
1 | 1111-1114 | Which of the given values of x and y make the following pair of matrices equal
3
7
5
1
2
3
x
y
x
+
+
−
,
0
2
8
4
y −
(A)
1,
7
3
x
y
=−
=
(B) Not possible to find
(C) y = 7,
32
x
=−
(D)
1
2
3,
3
x
y
−
−
=
=
10 The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:
(A) 27
(B) 18
(C) 81
(D) 512
3 4 Operations on Matrices
In this section, we shall introduce certain operations on matrices, namely, addition of
matrices, multiplication of a matrix by a scalar, difference and multiplication of matrices 3 |
1 | 1112-1115 | The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:
(A) 27
(B) 18
(C) 81
(D) 512
3 4 Operations on Matrices
In this section, we shall introduce certain operations on matrices, namely, addition of
matrices, multiplication of a matrix by a scalar, difference and multiplication of matrices 3 4 |
1 | 1113-1116 | 4 Operations on Matrices
In this section, we shall introduce certain operations on matrices, namely, addition of
matrices, multiplication of a matrix by a scalar, difference and multiplication of matrices 3 4 1 Addition of matrices
Suppose Fatima has two factories at places A and B |
1 | 1114-1117 | 3 4 1 Addition of matrices
Suppose Fatima has two factories at places A and B Each factory produces sport
shoes for boys and girls in three different price categories labelled 1, 2 and 3 |
1 | 1115-1118 | 4 1 Addition of matrices
Suppose Fatima has two factories at places A and B Each factory produces sport
shoes for boys and girls in three different price categories labelled 1, 2 and 3 The
quantities produced by each factory are represented as matrices given below:
Suppose Fatima wants to know the total production of sport shoes in each price
category |
1 | 1116-1119 | 1 Addition of matrices
Suppose Fatima has two factories at places A and B Each factory produces sport
shoes for boys and girls in three different price categories labelled 1, 2 and 3 The
quantities produced by each factory are represented as matrices given below:
Suppose Fatima wants to know the total production of sport shoes in each price
category Then the total production
In category 1 : for boys (80 + 90), for girls (60 + 50)
In category 2 : for boys (75 + 70), for girls (65 + 55)
In category 3 : for boys (90 + 75), for girls (85 + 75)
This can be represented in the matrix form as
80
90
60
50
75
70
65
55
90
75
85
75
+
+
+
+
+
+
|
1 | 1117-1120 | Each factory produces sport
shoes for boys and girls in three different price categories labelled 1, 2 and 3 The
quantities produced by each factory are represented as matrices given below:
Suppose Fatima wants to know the total production of sport shoes in each price
category Then the total production
In category 1 : for boys (80 + 90), for girls (60 + 50)
In category 2 : for boys (75 + 70), for girls (65 + 55)
In category 3 : for boys (90 + 75), for girls (85 + 75)
This can be represented in the matrix form as
80
90
60
50
75
70
65
55
90
75
85
75
+
+
+
+
+
+
Rationalised 2023-24
44
MATHEMATICS
This new matrix is the sum of the above two matrices |
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