##### VMC VIQ Scholarship Test

ApplyRegister for Vidyamandir Intellect Quest. Get Scholarship and Cash Rewards.

Edited By Kuldeep Maurya | Updated on Jan 20, 2022 02:54 PM IST

RD Sharma materials are considered the gold standard for CBSE maths. This is because they are some of the most informative, detailed, and widely used material. Before RD Sharma materials, many students and teachers had clear concepts and helped understand the subject from a basic level.

**Also Read - **RD Sharma Solution for Class 9 to 12 Maths

- Chapter 6 - Adjoint & Inverse of Matrix Ex 6.2
- Chapter 6 - Adjoint & Inverse of Matrix Ex FBQ
- Chapter 6 - Adjoint & Inverse of Matrix Ex MCQ
- Chapter 6 - Adjoint & Inverse of Matrix Ex VSA

**JEE Main 2025: Sample Papers | Mock Tests | PYQs | Study Plan 100 Days**

**JEE Main 2025: Maths Formulas | Study Materials**

**JEE Main 2025: Syllabus | Preparation Guide | High Scoring Topics **

Adjoint and Inverse of a Matrix Exercise 6.1 Question 1 (i)

Here, we use basic concept of adjoint of matrix .

Taking transpose,

(1)

(2)

(3)

From equation (1), (2) and (3)

Adjoint and Inverse of a Matrix Exercise 6.1 Question 1 (ii)

Here, we use basic concept of adjoint of matrix.

Let

Let’s find cofactor

Let’s transpose

Let’s prove this below

(1)

MHS

(2)

RHS

(3)

From equation (1), (2) and (3)

Adjoint and Inverse of a Matrix Exercise 6.1 Question 1 (iii)

Here, we use basic concept of determinant.

Let’s find

Let’s find cofactor

Let’s transpose

Let’s prove below

(1)

(2)

(3)

From equation (1), (2) and (3)

Adjoint and Inverse of a Matrix exercise 6.1 question 1 (iv)

Here, we use basic concept of determinant.

Let’s find

Let’s find cofactor

Take transpose of

Let’s prove below

(1)

(2)

(3)

From equation (1), (2) and 3

Hence proved

Adjoint and Inverse of a Matrix exercise 6.1 question 2 (i)

Here, we use basic concept of adjoint of matrix.

Let’s find

Let’s find cofactors

Let’s transpose it

Let’s verify below

(1)

(2)

(3)

From equation (1), (2) and (3)

Adjoint and Inverse of a Matrix exercise 6.1 question 2 (ii)

Here, we use basic concept of adjoint of matrix.

Let’s find

Let’s find cofactors

Let’s take transpose of

Let’s verify below

(1)

(2)

(3)

From equation (1), (2) and (3)

Adjoint and Inverse of Matrices Excercise 6.1 Question 2 (iii).**Answer:**

Here, we use basic concept of determinant and adjoint of matrix.

Let’s find

Let’s find cofactor

Let’s take transpose of

Let’s prove this below

(1)

(2)

(3)

So, from equation (1), (2) and (3)

Adjoint and Inverse of Matrices Excercise 6.1 Question 2 (iv).

Here, we use basic concept of determinant and adjoint of matrix.

Let’s find

Let’s find cofactor

Let’s take the transpose of

Let’s prove,

(1)

(2)

(3)

From the equation (1), (2) and (3)

Adjoint and Inverese of Matrices Exercise 6.1 Question 3

Proved

Here, we have to use advance method of finding adjoint of matrix.

We know that,

Then,

(1)

From the equation (1)

since,

So,

Adjoint and Inverese of Matrices Exercise 6.1 Question 4

Proved

Here, we use basic concept of adjoint of matrix.

Here, let’s find cofactor

So,

Let’s find

Take the transpose of

Hence we clearly see that

Adjoint and Inverese of Matrices Exercise 6.1 Question 5

Here, we use basic concept of determinant and adjoint of matrix.

Let’s find cofactors

So,

= Transpose of

(1)

(2)

Here, from equation (1) and (2)

Clearly see that,

Hence, proved

Adjoint and Inverse Matrix Exercise 6.1 Question 6

Here, we use basic concept of determinant.

Let’s find cofactor of A

= Transpose of

Hence,

Adjoint and Inverse Matrix Exercise 6.1 Question 7 (i)

Here, we use basic concept of determinant.

Let’s find cofactor of A

= Transpose of

Hence,

Adjoint and Inverse Matrix Exercise 6.1 Question 7 (ii)

Here, we use basic concept of inverse of matrix

Such that

Let’s find

So,

Is transpose of

So,

So,

Adjoint and Inverse Matrix Exercise 6.1 Question 7 (iii)

Here, we use basic concept of inverse.

We know that

Let’s find

So, let’s put value in formula

Adjoint and Inverse of a Matrix Exercise 6.1 Question 7 (i)

Here, we use basic concept of inverse

Let’s find

So,

Adjoint and Inverse of Matrices Excercise 6.1 Question 7 (iv)

Here, we use basic concept of determinant and inverse of matrix.

So let’s find

Let’s find

So,

Let’s put the values in formula

Adjoint and Inverse of Matrices Excercise 6.1 Question 8 (i).

Here, we use basic concept of determinant and inverse of matrix.

Let’s find

For that let’s find cofactor

Adjoint and Inverse of Matrices Excercise 6.1 Question 8 (ii)

Here, we use basic concept of determinant and inverse of matrix

We know that

So let’s find

Hence exist

Cofactor of A are

Adjoint and Inverse of Matrices Excercise 6.1 Question 8 (iii)

Here, we use basic concept of determinant and inverse of matrix.

Hence exist

Cofactor of A

Adjoint and Inverese of Matrices Exercise 6.1 Question 8 (iv)

.

Here, we use basic concept of determinant and inverse of matrix.

Hence exist

Cofactor of A are

Adjoint and Inverese of Matrices Exercise 6.1 Question 8 (v)

Here, we use basic concept of determinant and inverse of matrix.

Hence exist

Cofactor of A are

Adjoint and Inverese of Matrices Exercise 6.1 Question 8 (vi)

Here, we use basic concept of determinant and inverse of matrix

Hence exist

Cofactor of A are

Adjoint and Inverese of Matrices Exercise 6.1 Question 8 (vii)

Here, we use basic concept of determinant and inverse of matrix

Let’s find

So exist

Cofactors of A are

Adjoint and Inverse of a Matrix exercise 6.1 question 9 (i)

Here, we use basic concept of determinant and inverse of matrix.

Here, let’s find

Hence exist

Let’s find cofactor of A

Adjoint and Inverse of a Matrix exercise 6.1 question 9 (ii)

Here, we use basic concept of determinant and inverse of matrix

Hence exist

Cofactor of A

Hence

Adjoint and Inverse Matrix Exercise 6.1 Question 11

Here, we use basic concept of determinant and inverse of matrix

Let’s find

Now, we know that

Adjoint and Inverse Matrix Exercise 6.1 Question 12

Hence proved

Here, we use basic concept of determinant and inverse of matrix

Here let’s find

To show

(1)

(2)

From equation (1) and (2)

Hence,

Adjoint and Inverse Matrix Exercise 6.1 Question 13

Hence proved

Here, we use basic concept of determinant and inverse of matrix

Let’s find

To show

LHS

(1)

RHS

(2)

Here from equation (1) and (2)

Adjoint and Inverse of Matrices Excercise 6.1 Question 14

Here, we use basic concept of determinant and inverse of matrix

So, hence exist

Cofactor of A

To show

LHS

RHS

LHS = RHS

Adjoint and Inverse of Matrices Excercise 6.1 Question 15

Here, we use basic concept of determinant and inverse of matrix

For we know that

Here is also given so

Let’s fin

Cofactor of A are

Adjoint and Inverse of Matrices Excercise 6.1 Question 16 question (i)

Hence proved

Here, we use basic concept of determinant and inverse of matrix

Cofactor of A are

(1)

(2)

From equation (1) and (2)

Adjoint and Inverese of Matrices Exercise 6.1 Question 16 (ii)

Hence proved

Here, we use basic concept of determinant and inverse of matrix

Let’s find

Cofactor of A

Hence

Adjoint and Inverese of Matrices Exercise 6.1 Question 17

Here, we use basic concept of determinant and inverse of matrix

Hence

Let’s multiply both side

Adjoint and Inverse of Matrices Excercise 6.1 Question 18

Here, we use basic concept of determinant and inverse of matrix

Now,

Multiplying by both sides

So,

Adjoint and Inverse of a Matrix exercise 6.1 question 19

Here, we use basic concept of determinant and inverse of matrix

Show that

Now,

Multiply by both sides

Adjoint and Inverse of Matrices Excercise 6.1 Question 20

,

Here, we use basic concept of determinant and inverse of matrix

So,

So

Multiply by both sides,

Adjoint and Inverse of Matrices Excercise 6.1 Question 21

Λ =1,

Here, we use basic concept of determinant and inverse of matrix

Now,

Multiply both side

Adjoint and Inverse of Matrices Excercise 6.1 Question 22

Here, we use basic concept of determinant and inverse of matrix

Now,

Multiply both side

Adjoint and Inverese of Matrices Exercise 6.1 Question 23

Here, we use basic concept of determinant and inverse of matrix

Now,

Also,

Adjoint and Inverese of Matrices Exercise 6.1 Question 24

Here, we use basic concept of determinant and inverse of matrix

Now,

Now,

Now,

Hence,

Adjoint and Inverese of Matrices Exercise 6.1 Question 25

Here, we use basic concept of determinant and inverse of matrix

So,

Now,

Thus,

Adjoint and Inverse Matrix Exercise 6.1 Question 26**Answer:**

Here, we use basic concept of determinant and inverse of matrix

Now,

Thus,

Hence,

Adjoint and Inverse Matrix Exercise 6.1 Question 27

Here, we use basic concept of determinant and inverse of matrix

Find

Let’s find

Cofactor of A

Adjoint and Inverse Matrix Exercise 6.1 Question 28

Here, we use basic concept of determinant and inverse of matrix

RHS

Let’s find

Cofactor of A

(1)

RHS

(2)

So, here equation (1) and (2)

LHS = RHS

Adjoint and Inverse of Matrices Excercise 6.1 Question 29

Here, we use basic concept of determinant and inverse of matrix

Cofactor of A

Hence

Adjoint and Inverse of Matrices Excercise 6.1 Question 30

Here, we use basic concept of determinant and inverse of matrix

Cofactor of A

Adjoint and Inverse of a Matrix exercise 6.1 question 31

Here, we use basic concept of determinant and inverse of matrix

Cofactor of A

Adjoint and Inverse of a Matrix exercise 6.1 question 32

Here, we use basic concept of determinant and inverse of matrix

Then the given equation becomes as,

Adjoint and Inverse of a Matrix exercise 6.1 question 33

Here, we use basic concept of determinant and inverse of matrix

Then the given equation becomes

Adjoint and Inverse of a Matrix exercise 6.1 question 33

Edit Q

Adjoint and Inverse of a Matrix exercise 6.1 question 33

Here, we use basic concept of determinant and inverse of matrix

Then the given equation becomes

Adjoint and Inverese of Matrices Exercise 6 point 1 Question 34

Here, we use basic concept of determinant and inverse of matrix

Also,

Adjoint and Inverese of Matrices Exercise 6.1 Question 35

Here, we use basic concept of determinant and inverse of matrix

A is square matrix

Adjoint and Inverese of Matrices Exercise 6.1 Question 36

Here, we use basic concept of determinant and inverse of matrix

Find

So, we know that

So let’s find

Now cofactor of B

Now,

Adjoint and Inverse Matrix Exercise 6.1 Question 37

Here, we use basic concept of determinant and inverse of matrix

Let’s find

Cofactor of

Adjoint and Inverse Matrix Exercise 6.1 Question 38

Here, we use basic concept of determinant and inverse of matrix

Cofactor of A

Adjoint and Inverse Matrix Exercise 6.1 Question 39

Here, we use basic concept of determinant and inverse of matrix

Cofactor of A

Hence

Adjoint and Inverese of Matrices Exercise 6.1 Question 10 (i)

Proved

Here, we use basic concept of determinant and inverse of matrix

Let’s find

Then let’s find

Then find

Then let’s find and inverse of

(1)

Now

(2)

From equation (1) and (2)

Hence proved

Adjoint and Inverese of Matrices Exercise 6.1 Question 10 (ii)

Proved

Here, we use basic concept of determinant and inverse of matrix

Let’s find

Let’s find

Then find AB

Let’s find

(1)

Now

(2)

Hence proved from equation (1) and (2)

Adjoint and Inverese of Matrices Exercise 6.1 Question 16 (iii)

Hence proved

Here, we use basic concept of determinant and inverse of matrix

We have to show

We already know that,

Hence proved

RD Sharma Class 12th Exercise 6.1 consists of the chapter, Adjoint and Inverse of Matrix. This particular exercise consists of 58 Level 1 sums that are very fundamental and direct. Students can efficiently complete them in a day without a fuss if they understand the chapter. To help students cover as many questions as possible, Career360 has provided RD Sharma Class 12th Exercise 6.1 material.

The sums in this chapter are divided into two parts, i.e., Level 1 and Level 2. These levels are based on difficulty and weightage. Level one questions usually require fundamental knowledge and can be completed quickly, whereas level two sums require some extra understanding and are more complex.

It has solutions for the entire RD Sharma book that students can utilize to complete their syllabus. As it complies with the CBSE syllabus, students can refer to it for their classes and compare their progress. This exercise can be quickly completed with the help of RD Sharma Class 12th Exercise 6.1 by Career360.

Matrix multiplication, Adjoint, and Inverse, and other algebra are discussed in this chapter. As the material from Career360 contains solutions for all the questions from the book, there is nothing apart from this that students need to follow. These solutions are created by experts that have specifically designed them to help students get a good grasp of the subject. This is an easier and more efficient way to complete the preparation.

As RD Sharma Class 12th Exercise 6.1 is updated to the latest version, students don't have to worry about the differences. This is a simple one-stop-shop solution for all the exam needs when it comes to Math. RD Sharma's books contain a lot of questions that dive deep into the concepts.

Similarly, this chapter contains hundreds of questions. As teachers can't explain every one of those questions, this is where Career360 comes to help with RD Sharma Class 12 Chapter 6 Exercise 6.1 material. It has all the questions and covers the entire syllabus. It is beneficial and convenient for students now to study from home.

For the convenience of students, this material is free of cost. They can visit the website and download the material of choice for free. Thousands of students have already started preparing this material. Students who haven’t tried it yet should definitely refer to it.

- Chapter 1 - Relations
- Chapter 2 - Functions
- Chapter 3 - Inverse Trigonometric Functions
- Chapter 4 - Algebra of Matrices
- Chapter 5 - Determinants
- Chapter 6 - Adjoint and Inverse of a Matrix
- Chapter 7 - Solution of Simultaneous Linear Equations
- Chapter 8 - Continuity
- Chapter 9 - Differentiability
- Chapter 10 - Differentiation
- Chapter 11 - Higher Order Derivatives
- Chapter 12 - Derivative as a Rate Measurer
- Chapter 13 - Differentials, Errors and Approximations
- Chapter 14 - Mean Value Theorems
- Chapter 15 - Tangents and Normals
- Chapter 16 - Increasing and Decreasing Functions
- Chapter 17 - Maxima and Minima
- Chapter 18 - Indefinite Integrals
- Chapter 19 - Definite Integrals
- Chapter 20 - Areas of Bounded Regions
- Chapter 21 - Differential Equations
- Chapter 22 - Algebra of Vectors
- Chapter 23 - Scalar Or Dot Product
- Chapter 24 - Vector or Cross Product
- Chapter 25 - Scalar Triple Product
- Chapter 26 - Direction Cosines and Direction Ratios
- Chapter 27 - Straight Line in Space
- Chapter 28 - The Plane
- Chapter 29 - Linear programming
- Chapter 30- Probability
- Chapter 31 - Mean and Variance of a Random Variable

1. How can I download this material?

Students can download RD Sharma Class 12th Exercise 6.1 from Career360’s website for free. Simply search the book name and the exercise and download it on any laptop or mobile.

2. How are NCERT books compared to RD Sharma?

NCERT books are suitable for a basic study and contain some essential questions as well. Nevertheless, RD Sharma solutions are best suited for maths as they have detailed material. To check the material for matrices, check Class 12 RD Sharma Chapter 6 Exercise 6.1 Solution.

3. What is a Matrix?

A matrix is a collection of values of tabulated rows and columns. It is a rectangular array wherein the values are independent of each other and have no direct relation. To learn more about matrices, refer to RD Sharma Class 12 Solutions Adjoint and inverse of Matrix Ex 6.1.

4. What is the Inverse of a Matrix?

The inverse of a matrix is a value that, when multiplied, gives an identity matrix in multiple forms. If M is a matrix, then its inverse is denoted by M-1, and its equation can be given as M x M-1 = I. To learn more about the inverse of matrices, refer, RD Sharma Class 12 Solutions Adjoint and inverse of Matrix Ex 6.1.

5. Explain Adjoint of a Matrix

The adjoint is obtained by finding the cofactors and then finding the transpose of that resultant matrix. To learn more about adjoints of matrices, check RD Sharma Class 12 Solutions Chapter 6 Ex 6.1.

Get answers from students and experts

Register for Vidyamandir Intellect Quest. Get Scholarship and Cash Rewards.

Register for Tallentex '25 - One of The Biggest Talent Encouragement Exam

As per latest 2024 syllabus. Physics formulas, equations, & laws of class 11 & 12th chapters

As per latest 2024 syllabus. Chemistry formulas, equations, & laws of class 11 & 12th chapters

Accepted by more than 11,000 universities in over 150 countries worldwide

Register now for PTE & Unlock 10% OFF : Use promo code: 'C360SPL10'. Limited Period Offer!

News and Notifications

Back to top