RD Sharma Solutions For Class 12 Maths Exercise 7.1 Chapter 7 Adjoint and Inverse of a Matrix

The solutions for the exercise wise answers are prepared by experts at BYJU’S in the best possible way and are easily understandable by students. The PDF of RD Sharma Solutions for Class 12, Exercise 7.1 of Chapter 7 Adjoint and Inverse of a Matrix can be downloaded from the given links. This exercise consists of two levels according to the increasing order of difficulties. Let us have a look at important topics covered in this exercise.

  • Definition and meaning of adjoint of a square matrix
  • The inverse of a matrix
  • Some useful results on invertible matrices
  • Determining the adjoint and inverse of a matrix
  • Determining the inverse of a matrix when it satisfies the matrix equation
  • Finding the inverse of a matrix by using the definition of inverse
  • Finding a non – singular matrix when adjoint is given

RD Sharma Solutions For Class 12 Chapter 7 Adjoint and Inverse of a Matrix Exercise 7.1:-Download PDF Here

RD Sharma Class 12 Maths Solutions Chapter 7 Adjoint And Inverse Of A Matrix Exercise 7.1
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Exercise 7.2 Solutions

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Exercise 7.1 Page No: 7.22

1. Find the adjoint of each of the following matrices:

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 1
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 2
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 3
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 4

Verify that (adj A) A = |A| I = A (adj A) for the above matrices.

Solution:

(i) Let

A = 
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 5

Cofactors of A are

C11 = 4

C12 = – 2

C21 = – 5

C22 = – 3

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 6
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 7

(ii) Let

A = 
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 8

Therefore cofactors of A are

C11 = d

C12 = – c

C21 = – b

C22 = a

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 9
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 10

(iii) Let

A = 
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 11

Therefore cofactors of A are

C11 = cos α

C12 = – sin α

C21 = – sin α

C22 = cos α

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 12
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 13

(iv) Let

A = 
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 14

Therefore cofactors of A are

C11 = 1

C12 = tan α/2

C21 = – tan α/2

C22 = 1

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 15
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 16

2. Compute the adjoint of each of the following matrices.

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 17
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 18

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 19
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 20

Solution:

(i) Let

A = 
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 21

Therefore cofactors of A are

C11 = – 3

C21 = 2

C31 = 2

C12 = 2

C22 = – 3

C23 = 2

C13 = 2

C23 = 2

C33 = – 3

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 22
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 23

(ii) Let

A = 
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 24

Cofactors of A

C11 = 2

C21 = 3

C31 = – 13

C12 = – 3

C22 = 6

C32 = 9

C13 = 5

C23 = – 3

C33 = – 1

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 25
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 26
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 27

(iii) Let

A = 
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 28

Therefore cofactors of A

C11 = – 22

C21 = 11

C31 = – 11

C12 = 4

C22 = – 2

C32 = 2

C13 = 16

C23 = – 8

C33 = 8

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 29
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 30
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 31
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 32

(iv) Let

A = 
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 33

Therefore cofactors of A

C11 = 3

C21 = – 1

C31 = – 1

C12 = – 15

C22 = 7

C32 = – 5

C13 = 4

C23 = – 2

C33 = 2

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 34
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 35

Hence, (adj A) A = |A|I = A(adj A)

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 36

Solution:

Given

A = 
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 37

Therefore cofactors of A

C11 = 30

C21 = 12

C31 = – 3

C12 = – 20

C22 = – 8

C32 = 2

C13 = – 50

C23 = – 20

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 38
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 39
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 40

Solution:

Given

A = 
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 41

Cofactors of A

C11 = – 4

C21 = – 3

C31 = – 3

C12 = 1

C22 = 0

C32 = 1

C13 = 4

C23 = 4

C33 = 3

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 42
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 43

Solution:

Given

A = RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 44

Cofactors of A are

C11 = – 3

C21 = 6

C31 = 6

C12 = – 6

C22 = 3

C32 = – 6

C13 = – 6

C23 = – 6

C33 = 3

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 45
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 46

Solution:

Given

A = 
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 47

Cofactors of A are

C11 = 9

C21 = 19

C31 = – 4

C12 = 4

C22 = 14

C32 = 1

C13 = 8

C23 = 3

C33 = 2

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 48
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 49

7. Find the inverse of each of the following matrices:

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 50
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 51
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 52
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 53

Solution:

(i) The criteria of existence of inverse matrix is the determinant of a given matrix should not equal to zero.

Now, |A| = cos θ (cos θ) + sin θ (sin θ)

= 1

Hence, A – 1 exists.

Cofactors of A are

C11 = cos θ

C12 = sin θ

C21 = – sin θ

C22 = cos θ

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 54
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 55

(ii) The criteria of existence of inverse matrix is the determinant of a given matrix should not equal to zero.

Now, |A| = – 1 ≠ 0

Hence, A – 1 exists.

Cofactors of A are

C11 = 0

C12 = – 1

C21 = – 1

C22 = 0

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 56

(iii) The criteria of existence of inverse matrix is the determinant of a given matrix should not equal to zero.

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 57
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 58

(iv) The criteria of existence of inverse matrix is the determinant of a given matrix should not equal to zero.

Now, |A| = 2 + 15 = 17

Hence, A – 1 exists.

Cofactors of A are

C11 = 1

C12 = 3

C21 = – 5

C22 = 2

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 59

8. Find the inverse of each of the following matrices.

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 60
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 61
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 62
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 63
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 64
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 65
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 66

Solution:

(i) The criteria of existence of inverse matrix is the determinant of a given matrix should not equal to zero.

|A| = 
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 67

= 1(6 – 1) – 2(4 – 3) + 3(2 – 9)

= 5 – 2 – 21

= – 18

Hence, A – 1 exists

Cofactors of A are

C11 = 5

C21 = – 1

C31 = – 7

C12 = – 1

C22 = – 7

C32 = 5

C13 = – 7

C23 = 5

C33 = – 1

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 68
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 69

(ii) The criteria of existence of inverse matrix is the determinant of a given matrix should not equal to zero.

|A| = 
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 70

= 1 (1 + 3) – 2 (– 1 + 2) + 5 (3 + 2)

= 4 – 2 + 25

= 27

Hence, A – 1 exists

Cofactors of A are

C11 = 4

C21 = 17

C31 = 3

C12 = – 1

C22 = – 11

C32 = 6

C13 = 5

C23 = 1

C33 = – 3

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 71
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 72

(iii) The criteria of existence of inverse matrix is the determinant of a given matrix should not equal to zero.

|A| = 
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 73

= 2(4 – 1) + 1(– 2 + 1) + 1(1 – 2)

= 6 – 2

= – 4

Hence, A – 1 exists

Cofactors of A are

C11 = 3

C21 = 1

C31 = – 1

C12 = + 1

C22 = 3

C32 = 1

C13 = – 1

C23 = 1

C33 = 3

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 74

(iv) The criteria of existence of inverse matrix is the determinant of a given matrix should not equal to zero.

|A| = 
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 75

= 2(3 – 0) – 0 – 1(5)

= 6 – 5

= 1

Hence, A – 1 exists

Cofactors of A are

C11 = 3

C21 = – 1

C31 = 1

C12 = – 15

C22 = 6

C32 = – 5

C13 = – 5

C23 = – 2

C33 = 2

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 76

(v) The criteria of existence of inverse matrix is the determinant of a given matrix should not equal to zero.

|A| = 
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 77

= 0 – 1 (16 – 12) – 1 (– 12 + 9)

= – 4 + 3

= – 1

Hence, A – 1 exists

Cofactors of A are

C11 = 0

C21 = – 1

C31 = 1

C12 = – 4

C22 = 3

C32 = – 4

C13 = – 3

C23 = 3

C33 = – 4

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 78

(vi) The criteria of existence of inverse matrix is the determinant of a given matrix should not equal to zero.

|A| = 
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 79

= 0 – 0 – 1(– 12 + 8)

= 4

Hence, A – 1 exists

Cofactors of A are

C11 = – 8

C21 = 4

C31 = 4

C12 = 11

C22 = – 2

C32 = – 3

C13 = – 4

C23 = 0

C33 = 0

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 80

(vii) The criteria of existence of inverse matrix is the determinant of a given matrix should not equal to zero.

|A| = 
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 81 – 0 + 0

= – (cos2 α – sin2 α)

= – 1

Hence, A – 1 exists

Cofactors of A are

C11 = – 1

C21 = 0

C31 = 0

C12 = 0

C22 = – cos α

C32 = – sin α

C13 = 0

C23 = – sin α

C33 = cos α

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 82

9. Find the inverse of each of the following matrices and verify that A-1A = I3.

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 83
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 84

Solution:

(i) We have

|A| = 
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 85

= 1(16 – 9) – 3(4 – 3) + 3(3 – 4)

= 7 – 3 – 3

= 1

Hence, A – 1 exists

Cofactors of A are

C11 = 7

C21 = – 3

C31 = – 3

C12 = – 1

C22 = – 1

C32 = 0

C13 = – 1

C23 = 0

C33 = 1

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 86
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 87
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 88

(ii) We have

|A| = 
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 89

= 2(8 – 7) – 3(6 – 3) + 1(21 – 12)

= 2 – 9 + 9

= 2

Hence, A – 1 exists

Cofactors of A are

C11 = 1

C21 = 1

C31 = – 1

C12 = – 3

C22 = 1

C32 = 1

C13 = 9

C23 = – 5

C33 = – 1

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 90
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 91

10. For the following pair of matrices verify that (AB)-1 = B-1A-1.

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 92
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 93

Solution:

(i) Given

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 94
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 95

(ii) Given

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 96
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 97
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 98

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 99
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 100
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 101

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 102
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 103
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 104

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 105
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 106
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 107

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 108
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 109
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 110
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 111

Solution:

Given

A = 
https://gradeup-question-images.grdp.co/liveData/PROJ23872/1543574054891449.png and B – 1 = 
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 113

Here, (AB) – 1 = B – 1 A – 1

|A| = – 5 + 4 = – 1

Cofactors of A are

C11 = – 1

C21 = 8

C31 = – 12

C12 = 0

C22 = 1

C32 = – 2

C13 = 1

C23 = – 10

C33 = 15

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 114
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 115

(i) [F (α)]-1 = F (-α)

(ii) [G (β)]-1 = G (-β)

(iii) [F (α) G (β)]-1 = G (-β) F (-α)

Solution:

(i) Given

F (α) = 
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 116

|F (α)| = cos2 α + sin2 α = 1

Cofactors of A are

C11 = cos α

C21 = sin α

C31 = 0

C12 = – sin α

C22 = cos α

C32 = 0

C13 = 0

C23 = – 10

C33 = 1

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 117
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 118

(ii) We have

|G (β)| = cos2 β + sin2 β = 1

Cofactors of A are

C11 = cos β

C21 = sin α

C31 = sin β

C12 = 0

C22 = 1

C32 = 0

C13 = sin β

C23 = 0

C33 = cos β

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 119
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 120

(iii) Now we have to show that

[F (α) G (β)] – 1 = G (– β) F (– α)

We have already know that

[G (β)] – 1 = G (– β)

[F (α)] – 1 = F (– α)

And LHS = [F (α) G (β)] – 1

= [G (β)] – 1 [F (α)] – 1

= G (– β) F (– α)

Hence = RHS

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 121

Solution:

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 122
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 123
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 124

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 125
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 126
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 127
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 128

Solution:

Given

RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 129
RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix Image 130

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