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

RD Sharma Solutions for Class 12 Maths Exercise 7.1 Chapter 7 Adjoint and Inverse of a Matrix is provided here. The solutions for the exercise wise answers are prepared by the experts at BYJUâ€™S in the best possible way which 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 the 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 Maths Chapter 7 Adjoint and Inverse of a Matrix Exercise 7.1:-

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

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

Solution:

(i) Let

A =

Cofactors of A are

C11Â = 4

C12Â = â€“ 2

C21Â = â€“ 5

C22Â = â€“ 3

(ii) Let

A =

Therefore cofactors of A are

C11Â = d

C12Â = â€“ c

C21Â = â€“ b

C22Â = a

(iii) Let

A =

Therefore cofactors of A are

C11Â =Â cos Î±

C12Â =Â – sin Î±

C21Â =Â – sin Î±

C22Â =Â cos Î±

(iv) Let

A =

Therefore cofactors of A are

C11Â = 1

C12Â =Â tan Î±/2

C21Â =Â – tan Î±/2

C22Â = 1

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

Solution:

(i) Let

A =

Therefore cofactors of A are

C11Â = â€“ 3

C21Â = 2

C31Â = 2

C12Â = 2

C22Â = â€“ 3

C23Â = 2

C13Â = 2

C23Â = 2

C33Â = â€“ 3

(ii) Let

A =

Cofactors of A

C11Â = 2

C21Â = 3

C31Â = â€“ 13

C12Â = â€“ 3

C22Â = 6

C32Â = 9

C13Â = 5

C23Â = â€“ 3

C33Â = â€“ 1

(iii) Let

A =

Therefore cofactors of A

C11Â = â€“ 22

C21Â = 11

C31Â = â€“ 11

C12Â = 4

C22Â = â€“ 2

C32Â = 2

C13Â = 16

C23Â = â€“ 8

C33Â = 8

(iv) Let

A =

Therefore cofactors of A

C11Â = 3

C21Â = â€“ 1

C31Â = 1

C12Â = â€“ 15

C22Â = 7

C32Â = â€“ 5

C13Â = 4

C23Â = â€“ 2

C33Â = 2

Solution:

Given

A =

Therefore cofactors of A

C11Â = 30

C21Â = 12

C31Â = â€“ 3

C12Â = â€“ 20

C22Â = â€“ 8

C32Â = 2

C13Â = â€“ 50

C23Â = â€“ 20

C33 = 5

Solution:

Given

A =

Cofactors of A

C11Â = â€“ 4

C21Â = â€“ 3

C31Â = â€“ 3

C12Â = 1

C22Â = 0

C32Â = 1

C13Â = 4

C23Â = 4

C33Â = 3

Solution:

Given

A =

Cofactors of A are

C11Â = â€“ 3

C21Â = 6

C31Â = 6

C12Â = â€“ 6

C22Â = 3

C32Â = â€“ 6

C13Â = â€“ 6

C23Â = â€“ 6

C33Â = 3

Solution:

Given

A =

Cofactors of A are

C11Â = 9

C21Â = 19

C31Â = â€“ 4

C12Â = 4

C22Â = 14

C32Â = 1

C13Â = 8

C23Â = 3

C33Â = 2

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

Solution:

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

Now, |A| = cos Î¸Â (cos Î¸) + sinÂ Î¸Â (sinÂ Î¸)

= 1

Hence, AÂ â€“ 1Â exists.

Cofactors of A are

C11Â =Â cos Î¸

C12Â =Â sin Î¸

C21Â =Â – sin Î¸

C22Â =Â cos Î¸

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

Now, |A| = â€“ 1 â‰  0

Hence, AÂ â€“ 1Â exists.

Cofactors of A are

C11Â =Â 0

C12Â = â€“ 1

C21Â = â€“ 1

C22Â = 0

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

(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 â‰  0

Hence, AÂ â€“ 1Â exists.

Cofactors of A are

C11Â = 1

C12Â = 3

C21Â = â€“ 5

C22Â = 2

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

Solution:

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

|A| =

= 1(6 â€“ 1) â€“ 2(4 â€“ 3) + 3(2 â€“ 9)

= 5 â€“ 2 â€“ 21

= â€“ 18â‰  0

Hence, AÂ â€“ 1Â exists

Cofactors of A are

C11Â = 5

C21Â = â€“ 1

C31Â = â€“ 7

C12Â = â€“ 1

C22Â = â€“ 7

C32Â = 5

C13Â = â€“ 7

C23Â = 5

C33Â = â€“ 1

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

|A| =

= 1 (1 + 3) â€“ 2 (â€“ 1 + 2) + 5 (3 + 2)

= 4 â€“ 2 + 25

= 27â‰  0

Hence, AÂ â€“ 1Â exists

Cofactors of A are

C11Â = 4

C21Â = 17

C31Â = 3

C12Â = â€“ 1

C22Â = â€“ 11

C32Â = 6

C13Â = 5

C23Â = 1

C33Â = â€“ 3

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

|A| =

= 2(4 â€“ 1) + 1(â€“ 2 + 1) + 1(1 â€“ 2)

= 6 â€“ 2

= â€“ 4â‰  0

Hence, AÂ â€“ 1Â exists

Cofactors of A are

C11Â = 3

C21Â = 1

C31Â = â€“ 1

C12Â = + 1

C22Â = 3

C32Â = 1

C13Â = â€“ 1

C23Â = 1

C33Â = 3

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

|A| =

= 2(3 â€“ 0) â€“ 0 â€“ 1(5)

= 6 â€“ 5

= 1â‰  0

Hence, AÂ â€“ 1Â exists

Cofactors of A are

C11Â = 3

C21Â = â€“ 1

C31Â = 1

C12Â = â€“ 15

C22Â = 6

C32Â = â€“ 5

C13Â = 5

C23Â = â€“ 2

C33Â = 2

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

|A| =

= 0 â€“ 1 (16 â€“ 12) â€“ 1 (â€“ 12 + 9)

= â€“ 4 + 3

= â€“ 1â‰  0

Hence, AÂ â€“ 1Â exists

Cofactors of A are

C11Â = 0

C21Â = â€“ 1

C31Â = 1

C12Â = â€“ 4

C22Â = 3

C32Â = â€“ 4

C13Â = â€“ 3

C23Â = 3

C33Â = â€“ 4

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

|A| =

= 0 â€“ 0 â€“ 1(â€“ 12 + 8)

= 4â‰  0

Hence, AÂ â€“ 1Â exists

Cofactors of A are

C11Â = â€“ 8

C21Â = 4

C31Â = 4

C12Â = 11

C22Â = â€“ 2

C32Â = â€“ 3

C13Â = â€“ 4

C23Â = 0

C33Â = 0

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

|A| =
Â â€“ 0 + 0

= – (cos2 Î± â€“ sin2 Î±)

= â€“ 1â‰  0

Hence, AÂ â€“ 1Â exists

Cofactors of A are

C11Â = â€“ 1

C21Â = 0

C31Â = 0

C12Â = 0

C22Â =Â – cos Î±

C32Â =Â – sin Î±

C13Â = 0

C23Â =Â – sin Î±

C33Â =Â cos Î±

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

Solution:

(i) We have

|A| =

= 1(16 â€“ 9) â€“ 3(4 â€“ 3) + 3(3 â€“ 4)

= 7 â€“ 3 â€“ 3

= 1â‰  0

Hence, AÂ â€“ 1Â exists

Cofactors of A are

C11Â = 7

C21Â = â€“ 3

C31Â = â€“ 3

C12Â = â€“ 1

C22Â = 1

C32Â = 0

C13Â = â€“ 1

C23Â = 0

C33Â = 1

(ii) We have

|A| =

= 2(8 â€“ 7) â€“ 3(6 â€“ 3) + 1(21 â€“ 12)

= 2 â€“ 9 + 9

= 2â‰  0

Hence, AÂ â€“ 1Â exists

Cofactors of A are

C11Â = 1

C21Â = 1

C31Â = â€“ 1

C12Â = â€“ 3

C22Â = 1

C32Â = 1

C13Â = 9

C23Â = â€“ 5

C33Â = â€“ 1

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

Solution:

(i) Given

Hence, (AB)-1 = B-1A-1

(ii) Given

Hence, (AB)-1 = B-1A-1

Solution:

Given

Solution:

Given

Solution:

Given

Solution:

Solution:

Given

A =
Â and BÂ â€“ 1Â =

Here, (AB)Â â€“ 1 =Â BÂ â€“ 1Â AÂ â€“ 1

|A| = â€“ 5 + 4 = â€“ 1

Cofactors of A are

C11Â = â€“ 1

C21Â = 8

C31Â = â€“ 12

C12Â = 0

C22Â = 1

C32Â = â€“ 2

C13Â = 1

C23Â = â€“ 10

C33Â = 15

(i) [F (Î±)]-1 = F (-Î±)

(ii) [G (Î²)]-1 = G (-Î²)

(iii) [F (Î±) G (Î²)]-1 = G (-Î²) F (-Î±)

Solution:

(i) Given

F (Î±) =

|F (Î±)| =Â cos2 Î± + sin2Â Î± = 1â‰  0

Cofactors of A are

C11Â = cos Î±

C21Â = sin Î±

C31Â = 0

C12Â = â€“ sin Î±

C22Â = cos Î±

C32Â = 0

C13Â = 0

C23Â = 0

C33Â = 1

(ii) We have

|G (Î²)| =Â cos2 Î² + sin2Â Î²Â = 1

Cofactors of A are

C11Â = cos Î²

C21Â = 0

C31Â = -sin Î²

C12Â = 0

C22Â = 1

C32Â = 0

C13Â = sin Î²

C23Â = 0

C33Â = cos Î²

(iii) Now we have to show that

[F (Î±) G (Î²)]Â â€“ 1Â = G (â€“ Î²) F (â€“ Î±)

[G (Î²)]Â â€“ 1Â = G (â€“ Î²)

[F (Î±)]Â â€“ 1Â = F (â€“Â Î±)

And LHS =Â [F (Î±) G (Î²)]Â â€“ 1

=Â [G (Î²)]Â â€“ 1Â [F (Î±)]Â â€“ 1

=Â G (â€“ Î²) F (â€“ Î±)

Hence = RHS

Solution:

Consider,

Solution:

Given

Solution:

Given