# NCERT Solutions for Class 12 Maths Chapter 4 – Determinants

NCERT Solutions for Class 12 Maths Chapter 4 – Determinants are framed to help you prepare the chapter well for your class 12 board exams. This chapter tells you that Determinant of A order A determines the uniqueness of an NCERT Solutions of a matrix A associated with a system of linear equations. It also tells you about minors and cofactors, use of determinants in finding the area of a triangle, adjoint and inverse of a square matrix, consistency and inconsistency of system of linear equations and solution of linear equations in two or three variables using the inverse of a matrix.

NCERT Class 12 Maths Chapter 4 consists of 7 exercises containing 87 direct, multiple-choice, and skill-based problems asking you to evaluate determinants, prove equalities, write minors and cofactors etc. covering all the concepts learnt in the chapter.

All of these questions are covered in our smart and systematic NCERT Solutions for 12 Maths Chapter 4 – Determinants and have been** **answered using the best methods of solving these questions. These solutions are a hundred per cent accurate and effective study material. They give you an ample amount of practice so that you can excel in class 12^{th} board exams.

## NCERT Class 12 Maths Chapter 4

Our NCERT Solutions for Class 12 Maths Chapter 4-Determinants teach you many significant topics that are important from a board exam point of view. Some of them have been discussed below.

**Introduction**

This part of the NCERT CBSE Class 12 Maths Chapter 5 tells you the definition and meaning of determinants and its wide applications in science and humanitarian fields.

**Determinant**

This portion of the chapter gives you the idea of how linear algebraic equations can be converted into matrix form and to every square matrix, A = [aij] of order n. Determinants of a matrix could be of order 1, 2 or 3 × 3. This portion further teaches you how to expand a determinant along a row or a column in four steps, done them all together.

**Properties of Determinants**

This part of the CBSE NCERT Class 12 Maths Chapter 4 tells you about some properties of determinants which simplifies its evaluation by obtaining the maximum number of zeros in a row or a column.

You will learn here the following properties:

**Interchanging a matrix’s row and columns do not affect the value of the determinant.**

- If any two rows or columns are interchanged, the sign of the determinant of the given matrix is changed.

- In case any two rows or two columns are identical or proportional, then the determinant’s value is zero.

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

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

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

- If each element of a row or column of a matrix is zero, then the value of its determinant is also zero; if all elements on a certain side of primary diagonal are zeroes, the value if a determinant can be calculated by finding products of elements in the primary diagonal.

**Area of a Triangle**

This part of the NCERT Class 12 Maths Chapter 4 tells you that the area of a triangle can be calculated by plotting the coordinates of its vertices in a matrix form.

**Minors and Cofactors**

This part of the CBSE NCERT Class 12 Maths Chapter 4 teaches you the following

**Minor of a matrix can be found upon removing elements of a particular row, or a column.**

- Cofactor of a matrix is determined when a matrix’s elements are multiplied by specific signs.

- Adjoint and Inverse of a Matrix

This part of the NCERT CBSE Class 12 Maths Chapter 4 teaches you that the adjoint of a square matrix A = [aij]n × n is defined as the transpose of the matrix [Aij]n × n, where Aij is the cofactor of the element aij. Adjoint of the matrix A is denoted by adj A. The inverse of a matrix A can be written as A-1.

**Applications of Determinants and Matrices**

This chapter tells you that if there is one or more unique solution to a matrix, it is known as inconsistent and if no solution exists then that is an inconsistent matrix.

### NCERT CBSE Class 12 Maths Chapter 4 Exercises

Our NCERT Solutions for Class 12 Maths Chapter 4-Determinants answer all questions of all the 7 exercises of this chapter in a step by step manner to help you grasp all topics of this chapter easily and become thorough with the chapter. The components of the chapter have been discussed below.

**Exercise 4.1 **

This exercise consists of 8 questions. Questions 1, 2 and 5 ask you to evaluate the given determinants. Questions 3,4 and 6 ask you to find the determinant of A for the given matrices. Questions 7 and 8 ask you to find the value of unknown no. x for which the two given matrices are equal.

**Exercise 4.2**

This exercise consists of 14 questions. All the question from questions 1 to 14 asks you to use properties of determinants to prove equalities. Questions 1 to 7 does not need the determinants to be expanded. Questions 15 and 16 are multiple-choice type questions based on the concept learnt.

**Exercise 4.3**

This exercise contains 5 questions. Question 1 asks you to find the area of triangles using vertex points given. Questions 2 and 4 ask you to check if the given points are linear and to find equations for lines. Questions 3 and 5 ask you to determine the value of constant. *k.*

**Exercise 4.4**

This exercise contains 5 questions. Questions 1 and 2 ask you to write the minors and cofactors of the elements of given determinants. Questions 3 and 4 ask you to evaluate the determinants using cofactors of elements of indicated row/column. Question 5 is a multiple choice type question.

**Exercise 4.5**

This exercise comprises 18 questions. Questions 1 to 4 are based on adjoints of matrices. Questions 5 to 11 ask you to find the inverse of matrices given. Questions 12 to 16 are based on the concept of the inverse of matrix and questions 17 and 18 are multiple-choice questions based on the same concept.

**Exercise 4.6**

This exercise contains 16 questions. Questions 1 to 6 ask you to examine the inconsistency of the system of equations, and questions 7 to 14 ask you to solve the system of linear equations using matrix method. Question16 is a word problem. Question 15 asks you to solve a determinant using the inverse of the matrix.

**Exercise 4.7**

This exercise contains miscellaneous problems based on the concepts learnt in the chapter. It contains 19 questions. Questions 1 asks you to prove that determinant of trigonometric ratios is independent of angle θ. Questions 2, 4, 5 and then 11 to 15 ask you to prove equalities. Questions 3, 9 and 10 ask you to evaluate the given determinant. Questions 7 and 8 are based on the inverse of the matrix. Questions 17 to 19 are multiple-choice type questions.

### Benefits of NCERT Solutions for 12 Maths Chapter 4 – Determinants

Our NCERT Solutions for Class 12 Maths Chapter 4 – Determinants are the best study source for everyone studying for their class 12 board exams. Some of the benefits associated with learning and revising from our NCERT Solutions have been given below.

- They are based on the latest CBSE and NCERT curriculum.
- They entail all possible question types that can be asked in your board exams
- They give you ample practice to become thorough with the concepts of the chapter.
- They include detail explanations and relevant examples to help you learn efficiently.
- They help you in revising the contents thoroughly during exams and score more marks in class 10th board exams.
- Our NCERT Solutions have been carefully devised to make learning a more natural, enjoyable and happy process.