RD Sharma Solutions for Class 12 Maths Chapter 5 Algebra of Matrices

In order to have a good academic score in Mathematics, the important thing to be done by the students is to solve the questions of each and every exercise. RD Sharma Solutions for Class 12 is prepared by a team of experts who work with their full potential to help the students excel in their exams. Step by step solutions to each question in simple language, understandable by the students is the primary aim of the experts. The students can score good marks in the exams by solving all the questions and cross-checking the answers with the RD Sharma Solutions prepared by our faculty. RS Sharma Solutions for Class 12 Chapter 5 Algebra of Matrices PDF are given here. Let us have a look at some of the important concepts that are discussed in this chapter.

  • Definition and meaning of matrix
  • Types of matrices
  • Equality of matrices
  • Addition of matrices
  • Properties of matrix addition
  • Multiplication of a matrix by a scalar
  • Properties of scalar multiplication
  • Subtraction of matrices
  • Multiplication of matrices
  • Properties of matrix multiplication
  • Transpose of a matrix
  • Properties of transpose
  • Symmetric and skew-symmetric matrices

Download the PDF of RD Sharma Solutions For Class 12 Maths Chapter 5 Algebra of Matrices

 

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

Exercise 5.1 Page No: 5.6

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


Exercise 5.2 Page No: 5.18

1. Compute the following sums:

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 134

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 135

Solution:

(i) Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 136

Corresponding elements of two matrices should be added

Therefore, we get

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 137

Therefore,
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 138

(ii) Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 139

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 140

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 141

Therefore,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 142

RD Sharma Solutions for Class 12 Maths Chapter 5 Image143

Find each of the following:

(i) 2A – 3B

(ii) B – 4C

(iii) 3A – C

(iv) 3A – 2B + 3C

Solution:

(i) Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 144

First we have to compute 2A

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 145

Now by computing 3B we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 146

Now by we have to compute 2A – 3B we get

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 147

Therefore

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 148

(ii) Given
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 149

First we have to compute 4C,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 150

Now,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 151

Therefore we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 152

(iii) Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 153

First we have to compute 3A,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 154

Now,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 155

Therefore,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 156

(iv) Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 157

First we have to compute 3A

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 158

Now we have to compute 2B

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 159

By computing 3C we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 160

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 161

Therefore,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 162

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 163

(i) A + B and B + C

(ii) 2B + 3A and 3C – 4B

Solution:

(i) Consider A + B,

A + B is not possible because matrix A is an order of 2 x 2 and Matrix B is an order of 2 x 3, so the Sum of the matrix is only possible when their order is same.

Now consider B + C

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 164

(ii) Consider 2B + 3A

2B + 3A also does not exist because the order of matrix B and matrix A is different, so we cannot find the sum of these matrix.

Now consider 3C – 4B,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 165

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 166

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 167

Now we have to compute 2A – 3B + 4C

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 168

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 169

5. If A = diag (2 -5 9), B = diag (1 1 -4) and C = diag (-6 3 4), find

(i) A – 2B

(ii) B + C – 2A

(iii) 2A + 3B – 5C

Solution:

(i) Given A = diag (2 -5 9), B = diag (1 1 -4) and C = diag (-6 3 4)

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 170

(ii) Given A = diag (2 -5 9), B = diag (1 1 -4) and C = diag (-6 3 4)

We have to find B + C – 2A

Here,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 171

Now we have to compute B + C – 2A

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 172

(iii) Given A = diag (2 -5 9), B = diag (1 1 -4) and C = diag (-6 3 4)

Now we have to find 2A + 3B – 5C

Here,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 173

Now consider 2A + 3B – 5C

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 174

6. Given the matrices

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 175

Verify that (A + B) + C = A + (B + C)

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 176

Now we have to verify (A + B) + C = A + (B + C)

First consider LHS, (A + B) + C,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image177

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 178

Now consider RHS, that is A + (B + C)

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 179

Therefore LHS = RHS

Hence (A + B) + C = A + (B + C)

7. Find the matrices X and Y,

C:\Users\tnluser\Downloads\CodeCogsEqn (51).gif

Solution:

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 181

Now by simplifying we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 182

Therefore,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 183

Again consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 184

Now by simplifying we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 185

Therefore,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 186

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 187

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 188

Now by transposing, we get

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 189

Therefore,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 190

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 191

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 192

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 193

Now by multiplying equation (1) and (2) we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 194

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 195

Now by adding equation (2) and (3) we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 196

Now by substituting X in equation (2) we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 197

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 198

Solution:

Consider

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 199

Now, again consider

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 200

Therefore,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 201

And

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 202


Exercise 5.3 Page No: 5.41

1. Compute the indicated products:

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 203

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 204

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 205

Solution:

(i) Consider

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 206

On simplification we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 207

(ii) Consider

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 208

On simplification we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 209

(iii) Consider

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 210

On simplification we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 211

2. Show that AB ≠ BA in each of the following cases:

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 212

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 213

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 214

Solution:

(i) Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 215

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 216

Again consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 217

From equation (1) and (2), it is clear that

AB ≠ BA

(ii) Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 218

Now again consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 219

From equation (1) and (2), it is clear that

AB ≠ BA

(iii) Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 220

Now again consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 221

From equation (1) and (2), it is clear that

AB ≠ BA

3. Compute the products AB and BA whichever exists in each of the following cases:

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 222

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 223

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 224

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 225

Solution:

(i) Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 226

BA does not exist

Because the number of columns in B is greater than the rows in A

(ii) Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 227

Again consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 228

(iii) Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 229

AB = [0 + (-1) + 6 + 6]

AB = 11

Again consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 230

(iv) Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 231

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 232

4. Show that AB ≠ BA in each of the following cases:

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 233

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 234

Solution:

(i) Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 235

Again consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 236

From equation (1) and (2), it is clear that

AB ≠ BA

(ii) Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 237

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 238

Again consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 239

From equation (1) and (2) it is clear that,

AB ≠ BA

5. Evaluate the following:

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 240

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 241

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 242

Solution:

(i) Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 243

First we have to add first two matrix,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 244

On simplifying, we get

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 245

(ii) Given,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 246

First we have to multiply first two given matrix,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 247

= 82

(iii) Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 248

First we have subtract the matrix which is inside the bracket,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 249

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 250

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 251

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 252

We know that,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 253

Again we know that,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 254

Now, consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 255

We have,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 256

Now, from equation (1), (2), (3) and (4), it is clear that A2 = B2= C2= I2

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 257

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 258

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 259

Now we have to find,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 260

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 261

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 262

Solution:

Given

C:\Users\tnluser\Downloads\CodeCogsEqn (74).gif

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 264

Hence the proof.

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 265

Solution:

Given,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 266

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 267

Again consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 268

Hence the proof.

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 269

Solution:

Given,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 270

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 271

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 272

Hence the proof.

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 273

Solution:

Given,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 274

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 275

We know that,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 276

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 277

Again we have,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 278

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 279

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 280

Solution:

Given,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 281

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 282

Again consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 283

From equation (1) and (2) AB = BA = 03×3

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 284

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 285

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 286

Again consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 287

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 288

From equation (1) and (2) AB = BA = 03×3

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 289

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 290

Now consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 291

Therefore AB = A

Again consider, BA we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 292

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 293

Hence BA = B

Hence the proof.

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 294

Solution:

Given,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 295

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 296

Now again consider, B2

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 297

Now by subtracting equation (2) from equation (1) we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 298

16. For the following matrices verify the associativity of matrix multiplication i.e. (AB) C = A (BC)

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 299

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 300

Solution:

(i) Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 301

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 302

Now consider RHS,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 303

From equation (1) and (2), it is clear that (AB) C = A (BC)

(ii) Given,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 304

Consider the LHS,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 305

Now consider RHS,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 306

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 307

From equation (1) and (2), it is clear that (AB) C = A (BC)

17. For the following matrices verify the distributivity of matrix multiplication over matrix addition i.e. A (B + C) = AB + AC.

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 308

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 309

Solution:

(i) Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 310

Consider LHS,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 311

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 312

Now consider RHS,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 313

From equation (1) and (2), it is clear that A (B + C) = AB + AC

(ii) Given,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 314

Consider the LHS

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 315

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 316

Now consider RHS,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 317

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 318

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 319

Solution:

Given,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 320

Consider the LHS,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 321

Now consider RHS

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 322

From the above equations LHS = RHS

Therefore, A (B – C) = AB – AC.

19. Compute the elements a43 and a22 of the matrix:

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 323

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 324

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 325

From the above matrix, a43 = 8and a22 = 0

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 326

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 327

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 328

Again consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 329

Now, consider the RHS

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 330

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 331

Therefore, A3 = p I + q A + rA2

Hence the proof.

21. If ω is a complex cube root of unity, show that

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 332

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 333

It is also given that ω is a complex cube root of unity,

Consider the LHS,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 334

We know that 1 + ω + ω2 = 0 and ω3 = 1

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 335

Now by simplifying we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 336

Again by substituting 1 + ω + ω2 = 0 and ω3 = 1 in above matrix we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 337

Therefore LHS = RHS

Hence the proof.

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 338

Solution:

Given,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 339

Consider A2

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 340

Therefore A2 = A

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 341

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 342

Consider A2,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 343

Hence A2 = I3

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 344

Solution:

(i) Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 345

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 346

= [2x + 1 + 2 + x + 3] = 0

= [3x + 6] = 0

= 3x = -6

x = -6/3

x = -2

(ii) Given,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 347

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 348

On comparing the above matrix we get,

x = 13

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 349

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 350

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 351

⇒ [(2x + 4) x + 4 (x + 2) – 1(2x + 4)] = 0

⇒ 2x2 + 4x + 4x + 8 – 2x – 4 = 0

⇒ 2x2 + 6x + 4 = 0

⇒ 2x2 + 2x + 4x + 4 = 0

⇒ 2x (x + 1) + 4 (x + 1) = 0

⇒ (x + 1) (2x + 4) = 0

⇒ x = -1 or x = -2

Hence, x = -1 or x = -2

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 352

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 353

By multiplying we get,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 354

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 355

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 356

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 357

Now we have to prove A2 – A + 2 I = 0

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 358

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 359

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 360

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 361

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 362

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 363

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 364

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 365

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 366

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 367

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 368

Hence the proof.

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 369

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 370

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 371

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 372

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 373

Hence the proof.

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 374

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 375

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 376

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 377

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 378

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 379

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 380

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 381

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 382

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 383

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 384

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 385

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 386

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 387

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 388

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 389

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 390

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 391

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 392

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 393

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 394

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 395

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 396

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 397

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 398

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 399

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 400

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 401

I is identity matrix, so

 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 402

Also given, 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 403

Now, we have to find A2, we get

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 404

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 405

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 406

Now, we will find the matrix for 8A, we get

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 407

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 408

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 409

So,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 410

Substitute corresponding values from eqn (i) and (ii), we get

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 411

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 412

And to satisfy the above condition of equality, the corresponding entries of the matrices should be equal

Hence,

 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 413

Therefore, the value of k is 7

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 414

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 415

To show that f (A) = 0

Substitute x = A in f(x), we get

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 416

I is identity matrix, so 

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 417

Now, we will find the matrix for A2, we get

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 418

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 419

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 420

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 421

Now, we will find the matrix for 2A, we get

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 422

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 423

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 424

Substitute corresponding values from eqn (ii) and (iii) in eqn (i), we get

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 424

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 426

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 427

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 428

So,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 429

Hence Proved

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 430

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 431

So 

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 432

Now, we will find the matrix for A2, we get

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 433

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 434

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 435

Now, we will find the matrix for λ A, we get

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 436

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 437

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 438

But given, A2 = λ A + μ I

Substitute corresponding values from equation (i) and (ii), we get

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 439

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 440

And to satisfy the above condition of equality, the corresponding entries of the matrices should be equal

Hence, λ + 0 = 4 ⇒ λ = 4

And also, 2λ + μ = 7

Substituting the obtained value of λ in the above equation, we get

2(4) + μ = 7 ⇒ 8 + μ = 7 ⇒ μ = – 1

Therefore, the value of λ and μ are 4 and – 1 respectively

39. Find the value of x for which the matrix product

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 441

Solution:

We know,

 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 442

is identity matrix of size 3.

So according to the given criteria

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 443

Now we will multiply the two matrices on LHS using the formula cij = ai1b1j + ai2b2j + … + ain bnj, we get

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 444

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 445

And to satisfy the above condition of equality, the corresponding entries of the matrices should be equal

So we get

 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 446

So the value of x is 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 447


Exercise 5.4 Page No: 5.54

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 448

(i) (2A)T = 2 AT

(ii) (A + B)T = AT + BT

(iii) (A – B)T = AT – BT

(iv) (AB)T = BT AT

Solution:

(i) Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 449

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 450

Put the value of A

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 451

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 452

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 453

L.H.S = R.H.S

(ii) Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 454

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 455

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 456

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 457

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 458

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 459

L.H.S = R.H.S

Hence proved.

(iii) Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 460

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 461

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 462

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 463

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 464

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 465

L.H.S = R.H.S

(iv) Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 466

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 467

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 468

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 469

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 470

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 471

So,

 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 472

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 473

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 474

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 475

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 476

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 477

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 478

L.H.S = R.H.S

So, 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 479

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 480

(i) A + B)T = AT + BT

(ii) (AB)T = BT AT

(iii) (2A)T = 2 AT

Solution:

(i) Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 481

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 482

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 483

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 484

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 485

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 486

L.H.S = R.H.S

So, 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 487

(ii) Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 488

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 489

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 490

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 491

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 492

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 493

L.H.S = R.H.S

So, 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 494

(iii) Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 495

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 496

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 497

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 498

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 499

L.H.S = R.H.S

So,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 500

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 501

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 502

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 503

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 504

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 505

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 506

L.H.S = R.H.S

So,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 507

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 508

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 509

Now we have to find (AB)T

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 510

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 511

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 512

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 513

So,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 514


Exercise 5.5 Page No: 5.60

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 515

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 516

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 517

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 518

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 519

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 520 … (i)

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 521

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 522

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 523 … (ii)

From (i) and (ii) we can see that

A skew-symmetric matrix is a square matrix whose transpose equal to its negative, that is,

X = – XT

So, A – AT is a skew-symmetric.

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 524

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 525

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 526 … (i)

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 527

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 528

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 529 … (ii)

From (i) and (ii) we can see that

A skew-symmetric matrix is a square matrix whose transpose equals its negative, that is,

X = – XT

So, A – AT is a skew-symmetric matrix.

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 530

Solution:

Given,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 531 is a symmetric matrix.

We know that A = [aij]m × n is a symmetric matrix if aij = aji

So,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 532

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 533

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 534

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 535

Hence, x = 4, y = 2, t = -3 and z can have any value.

4. LetRD Sharma Solutions for Class 12 Maths Chapter 5 Image 536. Find matrices X and Y such that X + Y = A, where X is a symmetric and y is a skew-symmetric matrix.

Solution:

Given, 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 537 Then 
RD Sharma Solutions for Class 12 Maths Chapter 5 Image 538

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 539

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 540

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 541

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 542

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 543

Now,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 544

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 545

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 546

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 547

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 548

Now,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 549

X is a symmetric matrix.

Now,

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 550

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 551

-Y T = Y

Y is a skew symmetric matrix.

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 552

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 553

RD Sharma Solutions for Class 12 Maths Chapter 5 Image 554

Hence, X + Y = A


Also, access exercises of RD Sharma Solutions for Class 12 Maths Chapter 5 Algebra of Matrices

Exercise 5.1 Solutions

Exercise 5.2 Solutions

Exercise 5.3 Solutions

Exercise 5.4 Solutions

Exercise 5.5 Solutions

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