NCERT Solutions for Class 12 Maths Chapter 3 Exercise 3.3 Matrices - PDF Download

NCERT solutions for class 12 maths chapter 3 exercise 3.3 Matrices describes the topics such as transpose of a matrix, its properties and symmetric and skew symmetric matrices. Questions of exercise 3.3 are completely dependent on these topics. In ex 3.3 class 12 maths chapter 3, there are interactive examples to provide you a clear understanding of these topics. NCERT solutions class 12 maths chapter 3 ex 3.3 are prepared by highly qualified teachers of eSaral to make the complex concepts easy and accessible.

Class 12 maths chapter 3 exercise 3.3 NCERT solutions has a total of 12 questions that have been explained in step by step solutions to promote error-free learning. Ex 3.3 class 12 maths ch 3 solutions are also provided here in PDF format to help you in preparing for board exams. You can download the free PDF of ex 3.3 from the official website of eSaral and practice the questions in an easy way.

Topics Covered in Exercise 3.3 Class 12 Mathematics Questions

Ex 3.3 class 12 maths chapter 3 is completely based on transpose of a matrix and its properties, symmetric and skew symmetric matrices. You can find below the detailed explanation of these topics to get a clear understanding of exercise 3.3.

1. Transpose of a Matrix - If A = [a_ij] be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A is called the transpose of A. Transpose of the matrix A is denoted by A′ or (A^T ). In other words, if A = [a_ij] m × n , then A′ = [a_ji] n × m . 

If A  $=\left[\begin{array}{cc}3 & 5 \\ \sqrt{3} & 1 \\ 0 & \frac{-1}{5}\end{array}\right]_{3 \times 2}$ ,   then $\mathrm{A}^{\prime}=\left[\begin{array}{ccc}3 & \sqrt{3} & 0 \\ 5 & 1 & \frac{-1}{5}\end{array}\right]_{2 \times 3}$

• Properties of transpose of the matrices

The properties of the transpose of matrices are now stated without any proof. 

For any matrices A and B of suitable orders, we have

(i) (A′)′ = A,

(ii) (kA)′ = kA′ (where k is any constant)

(iii) (A + B)′ = A′ + B′

(iv) (A B)′ = B′ A′

2. Symmetric and Skew Symmetric Matrices

Symmetric matrices - A square matrix A = [a_ij] is said to be symmetric if A′ = A, that is, [a_ij] = [a_ji] for all possible values of i and j.

Skew Symmetric Matrices - A square matrix A = [a_ij] is said to be skew symmetric matrix if A′ = – A, that is a_ji = – a_ij for all possible values of i and j. Now, if we put i = j, we have a_ii = – a_ii. Therefore 2a_ii= 0 or a_ii = 0 for all i’s. 

This means that all the diagonal elements of a skew symmetric matrix are zero.

Theorem 1: For any square matrix A with real number entries, A + A′ is a symmetric matrix and A – A′ is a skew symmetric matrix.

Theorem 2: Any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix.

Tips for Solving Exercise 3.3 Class 12 Chapter 3 Matrices

NCERT solutions for class 12 maths chapter 3 ex 3.3 are effective resources that motivate students to practice more questions. You can also check here some of the useful tips to solve questions of ex 3.3.

1. Students should read all concepts and theorems included in NCERT solutions and solve all of the problems. Students can learn the appropriate methodology for conducting in-depth comprehension by practicing the questions thoroughly with the help of these solutions.

2. You should solve easy questions first then move on to the complex questions with step by step solutions.

3. NCERT solutions for ex 3.3 class 12 maths chapter 3 also has some essential theorems which you should learn to solve ex 3.3 questions.

Importance of Solving Ex 3.3 Class 12 Maths Chapter 3 Matrices

Ex 3.3 class 12 maths chapter 3 has a lot of benefits that will have a great impact while solving exercise questions. Here, our expert faculties of eSaral have combined some important benefits that you can check below.

1. NCERT solutions class 12 maths chapter 3 ex 3.3 are designed in descriptive format that will help you gain conceptual clarity of concepts to solve exercise questions.

2. These solutions of ex 3.3 include some significant theorems and properties of transpose of a matrix which are explained in a simple language to score good marks in exams.

3. By practicing questions in NCERT solutions class 12 maths chapter 3 ex 3.3 will help you improve your problem solving skills.

4. NCERT solutions PDF of ex 3.3 class 12 maths provides answers to all the questions that helps you to cross check your answers. 

Frequently Asked Questions

Question 1. What is a symmetric matrix?

Answer 1. A square matrix that is equal to its transpose is called symmetric. If A is a symmetric matrix then the following condition is satisfied: A = A^T

Question 2. What is a skew symmetric matrix?

Answer 2. A square matrix is said to be a skew symmetric matrix if the transpose is equal to its negative. It satisfies the condition  A′ = – A. 

Question 3. How can NCERT solutions for class 12 maths chapter 3 ex 3.3 help in preparing for exams?

Answer 3. When you are unable to solve ex 3.3 questions on your own, NCERT solutions for class 12 maths chapter 3 ex 3.3 will assist you in solving these difficulties. Example and sample questions included in NCERT solutions will be helpful for preparing for exams.