RD Sharma Solutions For Class 12 Maths Exercise 5.1 Chapter 5 Algebra of Matrices

This exercise has problems which are solved on Equality of Matrices of two matrices by the subject experts at BYJU’S. Our team of faculty mainly work with the aim of making Mathematics easier for students based on their grasping power. The students can gain a grip on the concepts covered in Maths Class 12 only by regular practise, and while solving can make use of the PDF as a reference. Solving the main problems will become easier if the students first solve the examples which are present before each exercise. RD Sharma Solutions for Class 12 Maths Chapter 5 Algebra of Matrices Exercise 5.1 are provided here.

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Access other exercises of RD Sharma Solutions For Class 12 Chapter 5 – Algebra of Matrices

Exercise 5.2 Solutions

Exercise 5.3 Solutions

Exercise 5.4 Solutions

Exercise 5.5 Solutions

Access answers to Maths RD Sharma Solutions For Class 12 Chapter 5 – Algebra of Matrices Exercise 5.1

1. If a matrix has 8 elements, what are the possible orders it can have? What if it has 5 elements?

Solution:

If a matrix is of order m × n elements, it has m n elements. So, if the matrix has 8 elements, we will find the ordered pairs m and n.

m n = 8

Then, ordered pairs m and n will be

m × n be (8 × 1),(1 × 8),(4 × 2),(2 × 4)

Now, if it has 5 elements

Possible orders are (5 × 1), (1 × 5).

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 1

Solution:

(i)

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 2

Now, Comparing with equation (1) and (2)

a22 = 4 and b21 = – 3

a22 + b21 = 4 + (– 3) = 1

(ii)

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 3

Now, Comparing with equation (1) and (2)

a11 = 2, a22 = 4, b11 = 2, b22 = 4

a11 b11 + a22 b22 = 2 × 2 + 4 × 4 = 4 + 16 = 20

3. Let A be a matrix of order 3 × 4. If R1 denotes the first row of A and C2 denotes its second column, then determine the orders of matrices R1 and C2.

Solution:

Given A be a matrix of order 3 × 4.

So, A = [ai j] 3×4

R1 = first row of A = [a11, a12, a13, a14]

So, order of matrix R1 = 1 × 4

C2 = second column of 

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 4

Therefore order of C2 = 3 × 1

4. Construct a 2 ×3 matrix A = [aj j] whose elements aj j are given by:

(i) ai j = i × j

(ii) ai j = 2i – j

(iii) ai j = i + j

(iv) ai j = (i + j)2/2

Solution:

(i) Given ai j = i × j

Let A = [ai j]2 × 3

So, the elements in a 2 × 3 matrix are

[a11, a12, a13, a21, a22, a23]

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 5

a11 = 1 × 1 = 1

a12 = 1 × 2 = 2

a13 = 1 × 3 = 3

a21 = 2 × 1 = 2

a22 = 2 × 2 = 4

a23 = 2 × 3 = 6

Substituting these values in matrix A we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 6

(ii) Given ai j = 2i – j

Let A = [ai j]2×3

So, the elements in a 2 × 3 matrix are

a11, a12, a13, a21, a22, a23

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 7

a11 = 2 × 1 – 1 = 2 – 1 = 1

a12 = 2 × 1 – 2 = 2 – 2 = 0

a13 = 2 × 1 – 3 = 2 – 3 = – 1

a21 = 2 × 2 – 1 = 4 – 1 = 3

a22 = 2 × 2 – 2 = 4 – 2 = 2

a23 = 2 × 2 – 3 = 4 – 3 = 1

Substituting these values in matrix A we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 8

(iii) Given ai j = i + j

Let A = [a i j] 2×3

So, the elements in a 2 × 3 matrix are

a11, a12, a13, a21, a22, a23

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 9

a11 = 1 + 1 = 2

a12 = 1 + 2 = 3

a13 = 1 + 3 = 4

a21 = 2 + 1 = 3

a22 = 2 + 2 = 4

a23 = 2 + 3 = 5

Substituting these values in matrix A we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 10

(iv) Given ai j = (i + j)2/2

Let A = [ai j]2×3

So, the elements in a 2 × 3 matrix are

a11, a12, a13, a21, a22, a23

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 11

Let A = [ai j]2×3

So, the elements in a 2 × 3 matrix are

a11, a12, a13, a21, a22, a23

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 12

a11 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 13

a12 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 14

a13 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 15

a21 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 16

a22 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 17

a23 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 18

Substituting these values in matrix A we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 19

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 20

5. Construct a 2 × 2 matrix A = [ai j] whose elements ai j are given by:

(i) (i + j)2 /2

(ii) ai j = (i – j)2 /2

(iii) ai j = (i – 2j)2 /2

(iv) ai j = (2i + j)2 /2

(v) ai j = |2i – 3j|/2

(vi) ai j = |-3i + j|/2

(vii) ai j = e2ix sin x j

Solution:

(i) Given (i + j)2 /2

Let A = [ai j]2×2

So, the elements in a 2 × 2 matrix are

a11, a12, a21, a22

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 21

a11 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 22

a12 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 23

a21 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 24

a22 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 25

Substituting these values in matrix A we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 26

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 27

(ii) Given ai j = (i – j)2 /2

Let A = [ai j]2×2

So, the elements in a 2 × 2 matrix are

a11, a12, a21, a22

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 28

a11 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 29

a12 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 30

a21 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 31

a22 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 32

Substituting these values in matrix A we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 33

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 34

(iii) Given ai j = (i – 2j)2 /2

Let A = [ai j]2×2

So, the elements in a 2 × 2 matrix are

a11, a12, a21, a22

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 35

a11 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 36

a12 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 37

a21 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 38

a22 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 39

Substituting these values in matrix A we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 40

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 41

(iv) Given ai j = (2i + j)2 /2

Let A = [ai j]2×2

So, the elements in a 2 × 2 matrix are

a11, a12, a21, a22

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 42

a11 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 43

a12 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 44

a21 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 45

a22 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 46

Substituting these values in matrix A we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 47

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 48

(v) Given ai j = |2i – 3j|/2

Let A = [ai j]2×2

So, the elements in a 2×2 matrix are

a11, a12, a21, a22

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 49

a11 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 50

a12 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 51

a21 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 52

a22 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 53

Substituting these values in matrix A we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 54

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 55

(vi) Given ai j = |-3i + j|/2

Let A = [ai j]2×2

So, the elements in a 2 × 2 matrix are

a11, a12, a21, a22

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 56

a11 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 57

a12 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 58

a21 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 59

a22 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 60

Substituting these values in matrix A we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 61

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 62

(vii) Given ai j = e2ix sin x j

Let A = [ai j]2×2

So, the elements in a 2 × 2 matrix are

a11, a12, a21, a22,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 63

a11 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 64

a12 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 65

a21 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 66

a22 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 67

Substituting these values in matrix A we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 68

6. Construct a 3×4 matrix A = [ai j] whose elements ai j are given by:
(i) ai j = i + j

(ii) ai j = i – j

(iii) ai j = 2i

(iv) ai j = j

(v) ai j = ½ |-3i + j|

Solution:

(i) Given ai j = i + j

Let A = [ai j]2×3

So, the elements in a 3 × 4 matrix are

a11, a12, a13, a14, a21, a22, a23, a24, a31, a32, a33, a34

A = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 69

a11 = 1 + 1 = 2

a12 = 1 + 2 = 3

a13 = 1 + 3 = 4

a14 = 1 + 4 = 5

a21 = 2 + 1 = 3

a22 = 2 + 2 = 4

a23 = 2 + 3 = 5

a24 = 2 + 4 = 6

a31 = 3 + 1 = 4

a32 = 3 + 2 = 5

a33 = 3 + 3 = 6

a34 = 3 + 4 = 7

Substituting these values in matrix A we get,

A = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 70

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 71


(ii) Given ai j = i – j

Let A = [ai j]2×3

So, the elements in a 3×4 matrix are

a11, a12, a13, a14, a21, a22, a23, a24, a31, a32, a33, a34

A = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 72

a11 = 1 – 1 = 0

a12 = 1 – 2 = – 1

a13 = 1 – 3 = – 2

a14 = 1 – 4 = – 3

a21 = 2 – 1 = 1

a22 = 2 – 2 = 0

a23 = 2 – 3 = – 1

a24 = 2 – 4 = – 2

a31 = 3 – 1 = 2

a32 = 3 – 2 = 1

a33 = 3 – 3 = 0

a34 = 3 – 4 = – 1

Substituting these values in matrix A we get,

A = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 73

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 74

(iii) Given ai j = 2i

Let A = [ai j]2×3

So, the elements in a 3×4 matrix are

a11, a12, a13, a14, a21, a22, a23, a24, a31, a32, a33, a34

A = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 75

a11 = 2×1 = 2

a12 = 2×1 = 2

a13 = 2×1 = 2

a14 = 2×1 = 2

a21 = 2×2 = 4

a22 = 2×2 = 4

a23 = 2×2 = 4

a24 = 2×2 = 4

a31 = 2×3 = 6

a32 = 2×3 = 6

a33 = 2×3 = 6

a34 = 2×3 = 6

Substituting these values in matrix A we get,

A = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 76

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 77

(iv) Given ai j = j

Let A = [ai j]2×3

So, the elements in a 3×4 matrix are

a11, a12, a13, a14, a21, a22, a23, a24, a31, a32, a33, a34

A = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 78

a11 = 1

a12 = 2

a13 = 3

a14 = 4

a21 = 1

a22 = 2

a23 = 3

a24 = 4

a31 = 1

a32 = 2

a33 = 3

a34 = 4

Substituting these values in matrix A we get,

A = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 79

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 80

(vi) Given ai j = ½ |-3i + j|

Let A = [ai j]2×3

So, the elements in a 3×4 matrix are

a11, a12, a13, a14, a21, a22, a23, a24, a31, a32, a33, a34

A = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 81

a11 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 82a12 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 83a13 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 84a14 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 85a21 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 86

a22 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 87

a23 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 88

a24 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 89a31 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 90a32 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 91a33 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 92a34 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 93

Substituting these values in matrix A we get,

A = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 94

Multiplying by negative sign we get,

7. Construct a 4 × 3 matrix A = [ai j] whose elements ai j are given by:

(i) ai j = 2i + i/j

(ii) ai j = (i – j)/ (i + j)

(iii) ai j = i

Solution:

(i) Given ai j = 2i + i/j

Let A = [ai j]4×3

So, the elements in a 4 × 3 matrix are

a11, a12, a13, a21, a22, a23, a31, a32, a33, a41, a42, a43

A = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 95

a11 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 96

a12 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 97

a13 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 98

a21 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 99

a22 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image100

a23 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 101

a31 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 102

a32 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 103

a33 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 104

a41 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 105

a42 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 106

a43 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 107

Substituting these values in matrix A we get,

A = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 108

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 109

(ii) Given ai j = (i – j)/ (i + j)

Let A = [ai j]4×3

So, the elements in a 4 × 3 matrix are

a11, a12, a13, a21, a22, a23, a31, a32, a33, a41, a42, a43

A = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 110

a11 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image111

a12 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 112

a13 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 113

a21 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 114

a22 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 115

a23 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 116

a31 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 117

a32 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 118

a33 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 119

a41 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 120

a42 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 121

a43 = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 122

Substituting these values in matrix A we get,

A = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 123

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 124

(iii) Given ai j = i

Let A = [ai j]4×3

So, the elements in a 4 × 3 matrix are

a11, a12, a13, a21, a22, a23, a31, a32, a33, a41, a42, a43

A = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 125

a11 = 1

a12 = 1

a13 = 1

a21 = 2

a22 = 2

a23 = 2

a31 = 3

a32 = 3

a33 = 3

a41 = 4

a42 = 4

a43 = 4

Substituting these values in matrix A we get,

A = 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 126

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 127

8. Find x, y, a and b if

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 128

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 129

Given that two matrices are equal.

We know that if two matrices are equal then the elements of each matrices are also equal.

Therefore by equating them we get,

3x + 4y = 2 …… (1)

x – 2y = 4 …… (2)

a + b = 5 …… (3)

2a – b = – 5 …… (4)

Multiplying equation (2) by 2 and adding to equation (1), we get

3x + 4y + 2x – 4y = 2 + 8

⇒ 5x = 10

⇒ x = 2

Now, substituting the value of x in equation (1)

3 × 2 + 4y = 2

⇒ 6 + 4y = 2

⇒ 4y = 2 – 6

⇒ 4y = – 4

⇒ y = – 1

Now by adding equation (3) and (4)

a + b + 2a – b = 5 + (– 5)

⇒ 3a = 5 – 5 = 0

⇒ a = 0

Now, again by substituting the value of a in equation (3), we get

0 + b = 5

⇒ b = 5

∴ a = 0, b = 5, x = 2 and y = – 1

9. Find x, y, a and b if

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 130

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 131

We know that if two matrices are equal then the elements of each matrices are also equal.

Given that two matrices are equal.

Therefore by equating them we get,

2a + b = 4 …… (1)

And a – 2b = – 3 …… (2)

And 5c – d = 11 …… (3)

4c + 3d = 24 …… (4)

Multiplying equation (1) by 2 and adding to equation (2)

4a + 2b + a – 2b = 8 – 3

⇒ 5a = 5

⇒ a = 1

Now, substituting the value of a in equation (1)

2 × 1 + b = 4

⇒ 2 + b = 4

⇒ b = 4 – 2

⇒ b = 2

Multiplying equation (3) by 3 and adding to equation (4)

15c – 3d + 4c + 3d = 33 + 24

⇒ 19c = 57

⇒ c = 3

Now, substituting the value of c in equation (4)

4 × 3 + 3d = 24

⇒ 12 + 3d = 24

⇒ 3d = 24 – 12

⇒ 3d = 12

⇒ d = 4

∴ a = 1, b = 2, c = 3 and d = 4

10. Find the values of a, b, c and d from the following equations:

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 132

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 133

We know that if two matrices are equal then the elements of each matrices are also equal.

Given that two matrices are equal.

Therefore by equating them we get,

2a + b = 4 …… (1)

And a – 2b = – 3 …… (2)

And 5c – d = 11 …… (3)

4c + 3d = 24 …… (4)

Multiplying equation (1) by 2 and adding to equation (2)

4a + 2b + a – 2b = 8 – 3

⇒ 5a = 5

⇒ a = 1

Now, substituting the value of a in equation (1)

2 × 1 + b = 4

⇒ 2 + b = 4

⇒ b = 4 – 2

⇒ b = 2

Multiplying equation (3) by 3 and adding to equation (4)

15c – 3d + 4c + 3d = 33 + 24

⇒ 19c = 57

⇒ c = 3

Now, substituting the value of c in equation (4)

4 × 3 + 3d = 24

⇒ 12 + 3d = 24

⇒ 3d = 24 – 12

⇒ 3d = 12

⇒ d = 4

∴ a = 1, b = 2, c = 3 and d = 4


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