## RD Sharma Solutions for Class 12 Maths Chapter 7 – Free PDF Download Updated for (2021-22)

**RD Sharma Solutions for Class 12 Maths Chapter 7 â€“ Adjoint and Inverse of a Matrix **is provided here. The RD Sharma textbook contains a huge number of solved examples and illustrations. It also provides quality content, easy stepwise explanations of various difficult concepts, and a wide variety of questions for practice. RD Sharma Solutions for Class 12 Chapter 7 are completely based on the exam-oriented approach to help the students in board exams. The PDF of RD Sharma Solutions for Class 12 Maths Chapter 7 Adjoint and Inverse of a Matrix is provided here.

Practising these **RD Sharma Solutions for Class 12** will ensure that the students can easily excel in their final examination of Mathematics. Students can refer to and download Chapter 7 Adjoint and Inverse of a Matrix from the given links. This chapter is based on the adjoint of a square matrix and its properties. RD Sharma Solutions cover all the topics related to it.

Some of the essential topics in **RD Sharma Solutions** of this chapter are listed below.

- 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
- Elementary transformation or elementary operations of a matrix
- Method of finding the inverse of a matrix by elementary transformation

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### Exercise 7.1 Page No: 7.22

**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

C_{11}Â = 4

C_{12}Â = â€“ 2

C_{21}Â = â€“ 5

C_{22}Â = â€“ 3

(ii) Let

A =

Therefore cofactors of A are

C_{11}Â = d

C_{12}Â = â€“ c

C_{21}Â = â€“ b

C_{22}Â = a

(iii) Let

A =

Therefore cofactors of A are

C_{11}Â =Â cos Î±

C_{12}Â =Â – sin Î±

C_{21}Â =Â – sin Î±

C_{22}Â =Â cos Î±

(iv) Let

A =

Therefore cofactors of A are

C_{11}Â = 1

C_{12}Â =Â tan Î±/2

C_{21}Â =Â – tan Î±/2

C_{22}Â = 1

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

** **

**Solution:**

(i) Let

A =

Therefore cofactors of A are

C_{11}Â = â€“ 3

C_{21}Â = 2

C_{31}Â = 2

C_{12}Â = 2

C_{22}Â = â€“ 3

C_{23}Â = 2

C_{13}Â = 2

C_{23}Â = 2

C_{33}Â = â€“ 3

(ii) Let

A =

Cofactors of A

C_{11}Â = 2

C_{21}Â = 3

C_{31}Â = â€“ 13

C_{12}Â = â€“ 3

C_{22}Â = 6

C_{32}Â = 9

C_{13}Â = 5

C_{23}Â = â€“ 3

C_{33}Â = â€“ 1

(iii) Let

A =

Therefore cofactors of A

C_{11}Â = â€“ 22

C_{21}Â = 11

C_{31}Â = â€“ 11

C_{12}Â = 4

C_{22}Â = â€“ 2

C_{32}Â = 2

C_{13}Â = 16

C_{23}Â = â€“ 8

C_{33}Â = 8

(iv) Let

A =

Therefore cofactors of A

C_{11}Â = 3

C_{21}Â = â€“ 1

C_{31}Â = 1

C_{12}Â = â€“ 15

C_{22}Â = 7

C_{32}Â = â€“ 5

C_{13}Â = 4

C_{23}Â = â€“ 2

C_{33}Â = 2

**Solution:**

Given

A =

Therefore cofactors of A

C_{11}Â = 30

C_{21}Â = 12

C_{31}Â = â€“ 3

C_{12}Â = â€“ 20

C_{22}Â = â€“ 8

C_{32}Â = 2

C_{13}Â = â€“ 50

C_{23}Â = â€“ 20

C_{33} = 5

**Solution:**

Given

A =

Cofactors of A

C_{11}Â = â€“ 4

C_{21}Â = â€“ 3

C_{31}Â = â€“ 3

C_{12}Â = 1

C_{22}Â = 0

C_{32}Â = 1

C_{13}Â = 4

C_{23}Â = 4

C_{33}Â = 3

**Solution:**

Given

A =

Cofactors of A are

C_{11}Â = â€“ 3

C_{21}Â = 6

C_{31}Â = 6

C_{12}Â = â€“ 6

C_{22}Â = 3

C_{32}Â = â€“ 6

C_{13}Â = â€“ 6

C_{23}Â = â€“ 6

C_{33}Â = 3

**Solution:**

Given

A =

Cofactors of A are

C_{11}Â = 9

C_{21}Â = 19

C_{31}Â = â€“ 4

C_{12}Â = 4

C_{22}Â = 14

C_{32}Â = 1

C_{13}Â = 8

C_{23}Â = 3

C_{33}Â = 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€“ 1}Â exists.

Cofactors of A are

C_{11}Â =Â cos Î¸

C_{12}Â =Â sin Î¸

C_{21}Â =Â – sin Î¸

C_{22}Â =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€“ 1}Â exists.

Cofactors of A are

C_{11}Â =Â 0

C_{12}Â = â€“ 1

C_{21}Â = â€“ 1

C_{22}Â = 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€“ 1}Â exists.

Cofactors of A are

C_{11}Â = 1

C_{12}Â = 3

C_{21}Â = â€“ 5

C_{22}Â = 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€“ 1}Â exists

Cofactors of A are

C_{11}Â = 5

C_{21}Â = â€“ 1

C_{31}Â = â€“ 7

C_{12}Â = â€“ 1

C_{22}Â = â€“ 7

C_{32}Â = 5

C_{13}Â = â€“ 7

C_{23}Â = 5

C_{33}Â = 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€“ 1}Â exists

Cofactors of A are

C_{11}Â = 4

C_{21}Â = 17

C_{31}Â = 3

C_{12}Â = â€“ 1

C_{22}Â = â€“ 11

C_{32}Â = 6

C_{13}Â = 5

C_{23}Â = 1

C_{33}Â = 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€“ 1}Â exists

Cofactors of A are

C_{11}Â = 3

C_{21}Â = 1

C_{31}Â = â€“ 1

C_{12}Â = + 1

C_{22}Â = 3

C_{32}Â = 1

C_{13}Â = â€“ 1

C_{23}Â = 1

C_{33}Â = 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€“ 1}Â exists

Cofactors of A are

C_{11}Â = 3

C_{21}Â = â€“ 1

C_{31}Â = 1

C_{12}Â = â€“ 15

C_{22}Â = 6

C_{32}Â = â€“ 5

C_{13}Â = 5

C_{23}Â = â€“ 2

C_{33}Â = 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€“ 1}Â exists

Cofactors of A are

C_{11}Â = 0

C_{21}Â = â€“ 1

C_{31}Â = 1

C_{12}Â = â€“ 4

C_{22}Â = 3

C_{32}Â = â€“ 4

C_{13}Â = â€“ 3

C_{23}Â = 3

C_{33}Â = 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€“ 1}Â exists

Cofactors of A are

C_{11}Â = â€“ 8

C_{21}Â = 4

C_{31}Â = 4

C_{12}Â = 11

C_{22}Â = â€“ 2

C_{32}Â = â€“ 3

C_{13}Â = â€“ 4

C_{23}Â = 0

C_{33}Â = 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

= – (cos^{2} Î± â€“ sin^{2} Î±)

= â€“ 1â‰ 0

Hence, AÂ ^{â€“ 1}Â exists

Cofactors of A are

C_{11}Â = â€“ 1

C_{21}Â = 0

C_{31}Â = 0

C_{12}Â = 0

C_{22}Â =Â – cos Î±

C_{32}Â =Â – sin Î±

C_{13}Â = 0

C_{23}Â =Â – sin Î±

C_{33}Â =Â cos Î±

**9. Find the inverse of each of the following matrices and verify that A ^{-1}A = I_{3}.**

**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€“ 1}Â exists

Cofactors of A are

C_{11}Â = 7

C_{21}Â = â€“ 3

C_{31}Â = â€“ 3

C_{12}Â = â€“ 1

C_{22}Â = 1

C_{32}Â = 0

C_{13}Â = â€“ 1

C_{23}Â = 0

C_{33}Â = 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€“ 1}Â exists

Cofactors of A are

C_{11}Â = 1

C_{21}Â = 1

C_{31}Â = â€“ 1

C_{12}Â = â€“ 3

C_{22}Â = 1

C_{32}Â = 1

C_{13}Â = 9

C_{23}Â = â€“ 5

C_{33}Â = â€“ 1

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

**Solution:**

(i) Given

Hence, (AB)^{-1} = B^{-1}A^{-1}

(ii) Given

Hence, (AB)^{-1} = B^{-1}A^{-1}

**Solution:**

Given

**Solution:**

Given

**Solution:**

Given

**Solution:**

**Solution:**

Given

A =

Â and BÂ ^{â€“ 1}Â =

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

|A| = â€“ 5 + 4 = â€“ 1

Cofactors of A are

C_{11}Â = â€“ 1

C_{21}Â = 8

C_{31}Â = â€“ 12

C_{12}Â = 0

C_{22}Â = 1

C_{32}Â = â€“ 2

C_{13}Â = 1

C_{23}Â = â€“ 10

C_{33}Â = 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 cos^{2} Î± + sin^{2}Â Î± = 1â‰ 0

Cofactors of A are

C_{11}Â = cos Î±

C_{21}Â = sin Î±

C_{31}Â = 0

C_{12}Â = â€“ sin Î±

C_{22}Â = cos Î±

C_{32}Â = 0

C_{13}Â = 0

C_{23}Â = 0

C_{33}Â = 1

(ii) We have

|G (Î²)| =Â cos^{2} Î² + sin^{2}Â Î²Â = 1

Cofactors of A are

C_{11}Â = cos Î²

C_{21}Â = 0

C_{31}Â = -sin Î²

C_{12}Â = 0

C_{22}Â = 1

C_{32}Â = 0

C_{13}Â = sin Î²

C_{23}Â = 0

C_{33}Â = cos Î²

(iii) Now we have to show that

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

We have already know that

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

^{â€“ 1}Â = F (â€“Â Î±)

And LHS =Â [F (Î±) G (Î²)]Â ^{â€“ 1}

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

=Â G (â€“ Î²) F (â€“ Î±)

Hence = RHS

**Solution:**

Consider,

**Solution:**

Given

**Solution:**

Given

### Exercise 7.2 Page No: 7.34

**Find the inverse of the following matrices by using elementary row transformations:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

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