**Class 12 Maths Chapter 3 Exercise 3.4 NCERT Solutions**

Class 12 Maths Exercise 3.4 Solutions

Class 12 Maths Exercise 3.4 (All Questions)

Class 12 Maths Exercise 3.4 in Hindi

Class 12 Maths Chapter 3 Solutions

Class 12 Maths NCERT Solutions

Class 12 all Subjects NCERT Solutions

Get the updated NCERT Solutions for Class 12 Maths Chapter 3 Exercise 3.4 Matrices in Hindi and English Medium modified and updated for session 2024-25. As per the syllabus only one question is there in class 12 Maths ex. 3.4 for academic year 2024-25.

## NCERT Solutions for Class 12 Maths Chapter 3 Exercise 3.4

Class: 12 | Mathematics |

Chapter 3: | Exercise 3.4 |

Topic: | Matrices |

Content Type: | Text and Videos Format |

Medium: | English and Hindi Medium |

### Class 12 Maths Exercise 3.4 Solutions in Hindi and English Medium

You can use 12th Maths Exercise 3.4 solutions online also without downloading. All the solutions of 12th Maths are updated for new academic session 2024-25. UP Board students are now officially using NCERT Books for course, so they can download UP Board solutions for class 12 Maths exercise 3.4 this page. All solutions are based on new CBSE Curriculum for CBSE Board, MP Board, UP Board and other boards students, who are following NCERT Books. Videos related to each question of Class 12 Maths Exercise 3.4 are just after the textbook solutions. Questions are describe properly including all minor steps, so that students can easily understand the sum.

#### 12th Maths Exercise 3.4 Solutions

NCERT Solutions for Class 12 Maths Chapter 3 Exercise 3.4 Matrices in English Medium update for the current academic session 2024-25 for UP Board and CBSE Board students who are following the latest NCERT Books for Class 12. Get all the exercises of Class 12 Mathematics Chapter 3 from main page. Share your knowledge with each other for NIOS board or CBSE Board in Discussion forum.

##### Important Questions with Answers for Practice

1. A matrix X has a + b rows and a + 2 columns while the matrix Y has b + 1 rows and a + 3 columns. Both matrices XY and YX exist. Find a and b. Can you say XY and YX are of the same type? Are they equal. [Answer: a = 2, b = 3, No]

2. Prove that the diagonal elements of a skew-symmetric matrix are all zero.

3. Give examples of matrices (i) A and B such that AB ≠ BA. (ii) A and B such that AB = O but A ≠ O, B ≠ O. (iii) A and B such that AB = O but BA ≠ O. (iv) A, B and C such that AB = AC but B ≠ C, A ≠ O.

4. If A and B are skew symmetric matrices of the same order, prove that AB + BA is symmetric matrix.

### Important Questions of 12th Maths Exercise 3.4

### Does the inverse of all Matrices exist?

In case, after applying one or more elementary row (column) operations on A = IA (A = AI), if we obtain all zeros in one or more rows of the matrix A on L.H.S., then A inverse does not exist.

### How do we operate Elementary Row Operations?

Let X, A and B be matrices of, the same order such that X = AB. In order to apply a sequence of elementary row operations on the matrix equation X = AB, we will apply these row operations simultaneously on X and on the first matrix A of the product AB on RHS.

### How do we operate Elementary Column Operations?

Let X, A and B be matrices of, the same order such that X = AB. In order to apply a sequence of elementary column operations on the matrix equation X = AB, we will apply, these operations simultaneously on X and on the second matrix B of the product AB on RHS.

### Prove that the inverse of a square matrix, if it exists, is unique.

Let A = [aij] be a square matrix of order m. If possible, let B and C be two inverses of A. We shall show that B = C.

Since B is the inverse of A

AB = BA = I … (1)

Since C is also the inverse of A

AC = CA = I … (2)

Thus,

B = BI = B (AC) = (BA) C = IC = C

###### 12th Maths Questions From Board Papers

- Let A and B be square matrices of the same order. Does (A + B)² = A² + 2AB + B² hold? If not, why?
- The monthly incomes of Mohan and Sohan are in the ratio 3:4 and their monthly expenditures are in the ratio 5:7. If each saves ₹15000 per month, find their monthly incomes and expenditures using matrices. [Answer: ₹90000 and ₹120000]
- For any square matrix write whether AA’ is symmetric of skew-symmetric.
- Three shopkeepers A, B and C go to a store to buy stationary. A purchase 12 dozen notebooks, 5 dozen pens and 6 dozen pencils. B purchases 10 dozen notebooks, 6 dozen pens and 7 dozen pencils. C purchases 11 dozen notebooks, 13 dozen pens and 8 dozen pencils. A notebook costs 40 paise, a pen costs ₹1.25 and a pencil costs 35 paise. Use matrix multiplication to calculate each individual’s bill. [Answer: ₹157.80, ₹167.40 and ₹281.40]

### What are the main points to study in exercise 3.4 of class 12th Maths?

The main points that students will study in exercise 3.4 of class 12th Maths are:

- Elementary operation (Transformation) of a matrix.

There are six operations (transformations) on a matrix, three of which are due to rows and three due to columns, which are known as elementary operations or transformations. - Invertible matrices.

If A is a square matrix of order m, and if there exists another square matrix B of the same order m, such that AB = BA = I, then B is called the inverse matrix of A and it is denoted by A⁻¹. In that case A is said to be invertible. - A rectangular matrix does not possess inverse matrix, since for products BA and AB to be defined and to be equal, it is necessary that matrices A and B should be square matrices of the same order.
- If B is the inverse of A, then A is also the inverse of B.
- The inverse of a matrix by elementary operations.

### Is exercise 3.4 of class 12th Maths important for the first term board exams?

Yes, exercise 3.4 of class 12th Maths is very important for the board exams. Every year questions come from this exercise in the board exams. 4 to 6 marks question comes from this exercise in the board exams. The most important problems of this exercise are examples 24, 25 and questions 8, 15, 16, 17.

### In which examples and questions of exercise 3.4 of 12th Maths, students face difficulty while solving?

Students face difficulty while solving example 24 and questions 15, 16, and 17 of exercise 3.4 of class 12th Maths. In these questions, students have to find the inverse of 3 × 3 matrices by using elementary operations (transformations). These questions require complete concentration.

### What is the approximate time needed to prepare exercise 3.4 of 12th mathematics?

Students need a maximum of 5 days to prepare exercise 3.4 (chapter 3) of class 12th mathematics if they give 2-3 hours per day to this exercise. This time also depends on student’s working speed, efficiency, capability, and many other factors.

### Which theorems are based on the inverse of a matrix exercise 3.4 of 12th Mathematics?

There are two theorems (theorems 3 and 4) that are based on the inverse of a matrix.

Theorem 3: (Uniqueness of inverse) Inverse of a square matrix, if it exists, is unique.

Theorem 4: If A and B are invertible matrices of the same order, then (AB)⁻¹ = B⁻¹ A⁻¹