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NCERT Solutions for Class 12 Maths Chapter 3 Matrices

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Matrices Class 12 NCERT Solutions Complete Resources - Free PDF Download

NCERT Class 12 Matrices Solutions PDF provided by Vedantu, offer detailed answers and explanations for all the exercises in this chapter. These solutions help students understand the concepts clearly and improve their problem-solving skills. Chapter 3 covers important topics that are crucial for exams, so focusing on understanding each concept is essential.

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Table of Content
1. Access Exercise Wise NCERT Solutions for Chapter 3 Maths Class 12
2. Master the Concepts of Matrices with NCERT Solution for Class 12 Maths Chapter 3
3. Some Important Points to remember
4. Summary of NCERT Solutions Class 12 Matrices 
5. Overview of Deleted Syllabus for Class 12 Maths Chapter 3
6. Class 12 Maths Chapter 3: Exercise Breakdown
7. Other Study Materials of CBSE Class 12 Maths Chapter 3
8. NCERT Solutions for Class 12 Maths | Chapter-wise List
9. Related Links for NCERT Class 12 Maths in Hindi
10. Important Related Links for NCERT Class 12 Maths
FAQs


In this chapter, key areas include various mathematical or scientific principles (depending on the subject), their applications, and problem-solving techniques. Students should pay special attention to the examples provided and practice the exercises regularly. The solutions by Vedantu are designed to make learning easier and more effective, ensuring students grasp the core ideas thoroughly.


Access Exercise Wise NCERT Solutions for Chapter 3 Maths Class 12

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Master the Concepts of Matrices with NCERT Solution for Class 12 Maths Chapter 3

NCERT Maths Class 12 Chapter 3 Solutions, "Matrices," is based on the concept of Matrix and their properties. The chapter consists of the following exercises:


Exercise 3.1: This exercise introduces types of matrices and covers basic operations like addition, subtraction, and multiplication.


Exercise 3.2: This exercise explores matrix properties like commutative, associative, and identity matrix use.


Exercise 3.3: This exercise focuses on matrix multiplication, its properties, and practicing inverse calculations.


Exercise 3.4: This exercise teaches and applies the concept of matrix transpose and its properties.


Miscellaneous Exercise: Mix of questions testing all matrix concepts and problem-solving skills.


Overall, this chapter is an important topic in linear algebra and covers the fundamental concepts of matrices, including matrix notation, matrix operations, matrix properties, and matrix multiplication.


Some Important Points to remember

1. A matrix is said to be an ordered rectangular array of numbers or functions. These numbers or functions in the array are called the elements or the entries of the matrix.


2. Order of a Matrix

The order of the matrix determines the dimension of the matrix and the number of rows and columns in the matrix. The general representation of matrix order is Amxn, where m is the number of rows and n is the number of columns in the matrix.


3. Types of Matrices

  • Column Matrix: A column matrix is a matrix with only one column.

  • Row Matrix: A row matrix is a matrix with only one row.

  • Square Matrix: A square matrix is one that has an equal number of rows and columns.

  • Diagonal Matrix: A square matrix where only the diagonal elements are non-zero and all other elements are zero.

  • Scalar Matrix: A diagonal matrix where all the diagonal elements are the same non-zero number.

  • Identity Matrix: A diagonal matrix where all diagonal elements are '1' and all other elements are zero, often symbolized as 'I'.

  • Zero or Null Matrix: A matrix where all elements are zero.


4. Equality of Matrices: Let A and B be two matrices. These matrices will be equal, if

(i) orders of A and B will be the same

(ii) corresponding elements of two matrices are the same


5. Operations on Matrices

  • Addition of Matrices: You can add two matrices by adding the numbers that are in the same position in each matrix, as long as both matrices are the same size.

  • Subtraction of Matrices: Subtracting matrices works like addition; you subtract the numbers in the same positions, but only if the matrices are the same size.

  • Multiplication of Matrices: To multiply matrices, the number of columns in the first matrix must match the number of rows in the second; you then multiply and sum specific pairs of numbers.


6. Properties of Multiplication of Matrices

  • Non-commutativity: 

Matrix multiplication will be not commutative i.e. if AB & BA are both defined, then it is not mandatory that AB ≠ BA.

  • Associative law:

For three matrices A, B, and C, if multiplication is defined, then we can write it as A (BC) = (AB) C.

  • Multiplicative identity: 

For any square matrix A, there will be an identity matrix of the same order in which  IA = AI = A.


Summary of NCERT Solutions Class 12 Matrices 

  • A matrix is an ordered rectangular array of numbers or functions.

  • A matrix having $m$ rows and $n$ columns is called a matrix of order $m \times n$.

  • $\left[a_y\right]_{m \times 1}$ is a column matrix.

  • $\left[a_h\right]_{\mathrm{b} x}$ is a row matrix.

  • An $m \times n$ matrix is a square matrix if $m=n$.

  • $A=\left[a_{i j}\right]_{\text {nex }}$ is a diagonal matrix if $a_{i j}=0$, when $i \neq j$.

  • $A_y=\left[a_{i j}\right]_{n \times n}$ is a scalar matrix if $a_{i j}=0$, when $i \neq j, a_{i j}=k,(k$ is some constant $)$, when $i=j$.

  • $A=\left[a_y\right]_{m \times n}$ is an identity matrix, if $a_{i j}=1$. when $i=j, a_{i j}=0$, when $i \neq j$.

  • A zero matrix has all its elements as zero.

  • $A=\left[a_i\right]-\left[b_y\right]-B$ if

  1. A and B are of asme order,

  2. $a_{i j}-b_{i j}$ for all possible values of $i$ and $j$.

  • $k a=k\left[a_y\right]_{m \times n}-\left[k\left(a_y\right)\right]_{m \times n}$

  • $-A=(-1) A$

  • $A-B=A+(-1) B$

  • $A+B=B+A$

  • $(A+B)+C=A+(B+C)$, where $A, B$ and $C$ are of aame order.

  • $k(A+B)=k A+k B$, where $\mathrm{A}$ and $\mathrm{B}$ of asme order, $k$ is constant.

  • $(k+l) A=k A+l A$, where $k$ and $l$ are constant.

  • If $A=\left[a_{j j}\right]_{m \beta n}$ and $B=\left[b_{i j}\right]_{N \times p}$, then $A B=C-\left[C_{j k}\right]_{w \times p}$, where $c_{j k}=\sum_{j=1}^n a_j b_{j k}$

  1. $A(B C)=(A B) C$

  2. $A(B+C)=A B+A C$

  3. $(A+B) C=A C+B C$

  •  If $A=\left[a_W\right]_{n \times N}$, then $A$ or $A^T=\left[a_N\right]_{N \times m}$

  1. $\left(A^{\prime}\right)^{\prime}=A$,

  2. $(k t)^{\prime}=k A^{\prime}$,

  3. $(A+B)^{\prime}=A^{\prime}+B^{\prime}$;

  4. $(A B)^{\prime}=B \cdot A^{\prime}$

  • $A$ is a symmetric matrix if $A^{\prime}=A$.

  • A is a skew symmetric matrix if $A^{\prime}=-A^{\prime}$.

  • Any aquare matrix can be represented as the sum of a symmetric and a skew symmetric matrix.

  • Elementary operations of a matrix are as follows:

  1. $R_i \leftrightarrow R_j$ or $C_i \leftrightarrow C_i$

  2. $R_l \rightarrow k R_i$ or $C_l \rightarrow k C_i$

  3. $R_i \rightarrow R_i+k R_j$ or $C_i \rightarrow C_i+k C_j$

  • If $A$ and $B$ re two square matrices such that $A B-B A=I$, then $B$ is the inverse matrix of $A$ and is denoted by $A^{-1}$ and $A$ is the inverse of $\mathrm{B}$.


Overview of Deleted Syllabus for Class 12 Maths Chapter 3

Chapter

Dropped Topics

Matrices

3.7 Elementary Operations (Transformation) of a Matrix

3.8.1 Inverse of Matrices by Elementary Operations (Retain Question 18 of Exercise 3.4)

Page 98 Example 26

Page Number 100-101: Miscellaneous Exercise Questions 1, 2, 3 and 12



Class 12 Maths Chapter 3: Exercise Breakdown

Exercise

Number of Questions

Exercise 3.1 Solutions

10 Questions (7 Short Answers, 3 MCQs)

Exercise 3.2 Solutions

22 Questions (14 Long, 6 Short, 2 MCQs)

Exercise 3.3 Solutions

12 Questions (10 Short Answers, 2 MCQs)

Exercise 3.4 Solutions

18 Questions (4 Long, 13 Short, 1 MCQ)

Miscellaneous Exercise Solutions

11 Questions and Solutions


Other Study Materials of CBSE Class 12 Maths Chapter 3



NCERT Solutions for Class 12 Maths | Chapter-wise List

Given below are the chapter-wise NCERT 12 Maths solutions PDF. Using these chapter-wise class 12th maths ncert solutions, you can get clear understanding of the concepts from all chapters.




Related Links for NCERT Class 12 Maths in Hindi

Explore these essential links for NCERT Class 12 Maths in Hindi, providing detailed solutions, explanations, and study resources to help students excel in their mathematics exams.




Important Related Links for NCERT Class 12 Maths

FAQs on NCERT Solutions for Class 12 Maths Chapter 3 Matrices

1. What are the main topics and subtopics of the chapter Matrices?

Matrices are one of the easiest chapters in maths, which, when understood, would become fun to solve. The concepts are as follows: Applications, Matrices, Order of matrices, Types of matrices includes Column matrix, Row matrix, Square matrix, Diagonal matrix, Scalar matrix, Identity matrix, Zero matrix and Equal matrix. You will also read about Operations of matrices which includes Addition of matrices, Subtraction of matrices and Multiplication of matrices. 


The chapter also explains about various laws which are: Commutative law, Associative law, Distributive law, Existence of multiplicative identity and Cancellation law. You will also get an idea of Transpose of a matrix, Properties of transpose, Symmetric and skew matrices and Invertible matrices.

2. Give me a glimpse of the chapter Matrices?

Matrices are majorly rectangular arrays of numbers which are represented in rows and columns. Basically, matrices are used to perform various mathematical operations such as addition, multiplication, subtraction and division. Representing data related to population, infant mortality rate, etc. are some of the widely used areas where matrices are implemented to simplify the calculations and ease the representation of data.


Matrices have substantial use in plotting graphs, statistics and various scientific research purposes. The most common and popular application of matrices is in solving linear equations. Matrices are even used to represent the coefficients of a linear equation. Other than that, matrices even find application in 3D maths, where they are used to define the relationship between two coordinate spaces.

3. How do our solutions help you in scoring good marks in the examination?

Start solving easier problems and slowly move to the medium level and then to the complex level. After each and every topic try to solve the questions and check where you are lagging behind. Practice the weaker area of that particular concept as many times as you can. This also helps you in knowing your strength along with the weakness. 


Maths is always a part of our life and it will also be used in our day to day life. The short-cut technique to score well in maths is to practice. Do not mug up the solutions, try to understand them and solve it in your own way and then check where you have gone wrong.

4. Why should we learn about Matrices in NCERT Solutions for Class 12 Maths Chapter 3?

Matrices are a powerful and essential tool in Mathematics. They are rectangular arrays of numbers represented in the form of rows and columns. It is used to perform several mathematical operations like addition, subtraction, multiplication, and division. It is widely used in various areas to simplify complex operations like plotting graphs, representing the population, statistics, and in various research papers. It also simplifies the method of solving complex linear equations. 

5. How many exercises are there in the Matrix?

There are four exercises and one miscellaneous exercise in the chapter Matrix. You can refer to the Maths Solutions for Class 12 by Vedantu, where you will find the detailed solution of every question in your Maths book. The solutions are explained in a simple and detailed manner and will clear your doubts. It also includes tips and important points that will help you score high in your exams. 

6. Is Class 12 maths tough?

No, Class 12 Maths is not tough. You can score high by regular practice and strong concepts. Solve every question in your NCERT Maths book, including the solved questions. Refer to Vedantu’s Maths Solution for Class 12. It focuses on strengthening your concepts so that you can solve a variety of questions. Practice some questions daily to improve your solving skills. You can refer to Vedantu’s Revision Class 12 Notes which includes all the important concepts and formulas compiled in one place. 

7. What are Matrices in Maths Class 12?

Matrices are rectangular arrays where data is represented in the form of rows and columns. It is a very easy and scoring topic. This chapter includes:

  • Types of Matrices

  • Operations on Matrices

  • Transpose of Matrices

  • Symmetric and Skew-symmetric Matrices

  • Elementary Operations (Transformation) on Matrices

  • Invertible Matrices

These topics are explained in easy language so that the students can score high. You can practice a variety of questions in this chapter since it is relatively easier. 

8. Why should you refer to Vedantu’s Solutions for Class 12 Maths?

Vedantu’s Solutions for Class 12 Maths are prepared by experienced subject-matter experts and include answers to every question in your NCERT Maths book. It includes all the important formulas, concepts, and theorems that are very important from the exam point of view. The solutions are updated according to the latest guidelines of the CBSE board. This is also very useful for revision before your exam. It includes step-wise solutions that will help you to understand the concepts well. The solution PDFs and other study materials such as important questions and revision notes can also be downloaded from the Vedantu app as well for free of cost.

9. Why understanding Metrices is crucial for exams in Class 12?

Mastery of matrices is essential for solving complex mathematical problems and is frequently tested in board exams.

10. What should I focus on in Chapter 3 Maths Class 12?

The NCERT Solutions Class 12 Maths Chapter 3 covers :

  • The order of a matrix

  • Types of matrices

  • Mathematical operation 

  • Transpose of a matrix

  • Symmetric matrices

  • Skew-symmetric matrices

  • Simple operations 

  • Invertible matrices.

11. How many questions from matrices appear in the board exams?

This chapter has 62 questions in 4 exercises along with 15 more provided in a miscellaneous exercise. Out of these 41 questions are short answer types, 11 multiple choice questions, and 25 long answer type questions.

12. Where can I find step-by-step process for matrices class 12 ncert solutions?

Vedantu provides comprehensive solutions that break down complex matrix problems into simple steps.

13. What are the trickiest topics in Chapter 3 Maths Class 12?

Focus more on matrix multiplication and finding inverses as these areas are often challenging for students.

14. How can I effectively practice matrix operations from Chapter 3?

Regular practice using NCERT exercises and additional problems on platforms like Vedantu is recommended.

15. Are matrix concepts important for other competitive exams too?

Yes, understanding matrices is useful for various competitive exams, especially in technical fields.