# NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.5

NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.5 Determinants in Hindi Medium as well as English Medium. Now NCERT Books are being used as textbooks for all the students of CBSE, Uttarakhand, Bihar board, MP Board, UP Board and other board also. So, these solutions are useful not only for CBSE board but UP Board, MP board and all other board who are using NCERT Books. Djownload here the UP Board solutions for Class 12 Maths Chapter 4 Exercise 4.5 in Hindi Medium and English Medium. Solutions are available in PDF format as well as Online study (without downloading). Videos related to all questions of Exercise 4.5

5 of 12th Maths are given below in Hindi and English Medium. Download NCERT Solutions for 2021-22 in updated form for new academic session. Offline apps work without internet once downloaded.## NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.5

### Class 12 Maths Exercise 4.5 Solutions in Hindi & English

Class: 12 | Maths (English and Hindi Medium) |

Chapter 4: | Exercise 4.5 |

### 12th Maths Exercise 4.5 Solutions

NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.5 Determinants English and Hindi Medium free to download for offline use. UP Board Solutions for Class 12 Maths Exercise 4.5 are also same as CBSE Board solutions. In this exercise, you have to learn how to find adjoint of a matrix A and how to find inverse of a square matrix. Download NCERT Books for Class 12 for Academic Session 2021-2022 for offline use.

#### Class 12 Maths Chapter 4 Exercise 4.5 Solution in Videos

#### Class 12 Maths Exercise 4.5 Solution in Hindi Videos

#### Class 12 Maths Exercise 4.5 Question 17, 18 Solution

#### Adjoint and Inverse of a Matrix

Adjoint and Inverse of a Matrix is important in finding the solutions of system of linear equation in many variables. During the researches in any field the linear equations of so many variables are formed, then Matrices helps to solve them easily. Adjoint of a matrix of order two can be directly by interchanging a11 & a11 and then changing the sign of a12 & a21.

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### Important Questions of 12th Maths Exercise 4.5

##### Where does one use determinants in real life?

Determinants can be used in mensuration to find areas and volumes. It is frequently used in research to solve linear equations. Determinants are also used in applied Mathematics.

##### Who initiate the using determinants as a tool in Mathematics?

The Chinese method of representing the coefficients of the unknowns of several linear equations by using rods on a calculating board naturally led to the discovery of simple method of elimination. The arrangement of rods was precisely that of the numbers in a determinant. The Chinese, therefore, early developed the idea of subtracting columns and rows as in simplification of a determinant Mikami, China, pp 30, 93.

##### What are the main properties of determinants?

The main properties of determinants are as follows:

1. If we interchange any two rows (or columns), then sign of determinant changes.

2. If any two rows or any two columns are identical or proportional, then value of determinant is zero.

3. If we multiply each element of a row or a column of a determinant by constant k, then value of determinant is multiplied by k.

4. Multiplying a determinant by k means multiply elements of only one row (or one column) by k.

5. If elements of a row or a column in a determinant can be expressed as sum of two or more elements, then the given determinant can be expressed as sum of two or more determinants.

6. If to each element of a row or a column of a determinant the equimultiples of corresponding elements of other rows or columns are added, then value of determinant remains same.

##### What does a determinant of 1 mean?

When the value of a determinant is 1, the matrix associated to this determinant is said to be unimodular.

##### Which theorems should students know before starting exercise 4.5 of grade 12th Maths?

Before starting exercise 4.5 of grade 12th Maths, students should know the following theorems:

- 1. Theorem 1: If A be any given square matrix of order n, then

A(adj A)=(adj A) A=|A| I , where I is the identity matrix of order n. - 2. Theorem 2: If A and B are nonsingular matrices of the same order, then AB and BA are also nonsingular matrices of the same order.
- 3. Theorem 3: The determinant of the product of matrices is equal to product of their respective determinants, i.e., |AB| = |A| |B|, where A and B are square matrices of the same order.
- 4. Theorem 4: A square matrix A is invertible if and only if A is nonsingular matrix.

##### Which questions and examples of exercise 4.5 of class 12th Maths are important from an exam point of view?

Exercise 4.5 has four examples (examples 23, 24, 25, 26) and 18 questions. Students should practice all questions and examples of this exercise for the exam. The most important examples and questions of this exercise are examples 24, 25, 26, and questions 3, 4, 6, 10, 11, 12, 13, 14, 15, 16, 17, 18.

##### When a square matrix is known as non-singular in exercise 4.5 of class 12th Maths?

1. A square matrix A is said to be singular if |A| = 0.

2. A square matrix A is said to be non-singular if |A| ≠ 0.

##### Can students complete exercise 4.5 of class 12th Maths within three days?

Students can complete exercise 4.5 of class 12th Maths within three days or not depends on student’s working speed, efficiency, capability, and many other factors. But if they try and give 2 hours per day to exercise 4.5 of class 12th Maths, they can complete it within three days.