NCERT Solutions For Class 12 Maths Chapter 4 Determinants are available at CoolGyan’S which are given by our expert faculty. The step by step solutions for all the questions enlisted under the chapter and each solution has been explained using suitable examples or formulas. Maths Class 12 NCERT Solutions are written by considering the marking scheme. CoolGyan’S aims to bring out the very best in students through additional skill-building exercises that are tailored to their grade levels, abilities and interests.
A determinant is a number that can be calculated for only square matrices. In solving a system of linear equations, determinant plays an important role to check whether the system of equations has a unique solution or not. It has many applications in science, engineering, social sciences, and economics, etc., NCERT Solutions of this chapter of Class 12 cover all the problems such as finding the determinant of a 2×2 matrix, 3×3 matrix, evaluating the area of the triangle, finding the equation of the line using determinant, minors, and co-factors. Also, calculating the determinant using minor and co-factor, adjoint, and the inverse of a matrix. Besides, finding the solution of equations and determinant as the sum of two or more determinants.
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NCERT Solutions for Class 12 Maths Chapter 4 Determinants
Chapter Determinants has a total of 6 exercises and a miscellaneous exercise which covers all the questions related to the chapter. The topics discussed in Chapter 4 NCERT Solutions For Class 12 of Maths includes:
4.1 Introduction
Students have already read about how to represent a system of linear equations using matrices and the condition to know whether this system has a unique solution or not based on the determinant. These things will be revised under this section.
4.2 Determinant
4.2.1 Determinant of a matrix of order one
4.2.2 Determinant of a matrix of order two
4.2.3 Determinant of a matrix of order 3 × 3
In this section, students will learn how to find the determinant of a square matrix of different orders such as one, two and three along with examples.
4.3 Properties of Determinants
In the previous section, students have learnt how to expand the determinants. In this section, they will study some properties of determinants which simplifies its evaluation by obtaining the maximum number of zeros in a row or a column. These properties are correct for determinants of any order. However, in this chapter, it has been limited up to determinants of order three only.
4.4 Area of a Triangle
We can find the area of a triangle using the formula when the coordinates of three vertices have been given. In this section, you will learn how to find the area of the triangle by converting the points in the form of a determinant.
4.5 Minors and Cofactors
Students will learn to write the expansion of a determinant in compact form using minors and cofactors after practicing the problems in this section.
4.6 Adjoint and Inverse of a Matrix
4.6.1 Adjoint of a matrix
After solving the problems in this section, you will understand clearly how to find the inverse of a matrix using adjoint. Many theorems and examples are given under this section to enhance your skills.
4.7 Applications of Determinants and Matrices
4.7.1 Solution of system of linear equations using inverse of a matrix
Here, you will get a complete description of applications of determinants and matrices for solving the system of linear equations in two or three variables and how to check the consistency of the system of linear equations.
Exercise 4.1 Solutions: 8 Questions (2 Long, 5 Short Answers, 1 MCQ)
Exercise 4.2 Solutions: 16 Questions(7 Long, 7 Short, 2 MCQs)
Exercise 4.3 Solutions: 5 Questions ( 4 Short Answers, 1 MCQ)
Exercise 4.4 Solutions: 5 Questions (4 Long, 1 MCQ)
Exercise 4.5 Solutions: 18 Questions (11 Long, 5 Short, 2 MCQs)
Exercise 4.6 Solutions: 16 Questions (13 Long, 3 Short)
Miscellaneous Exercise Solutions: 19 Questions (15 Long, 1 Short, 3 MCQs)
Key Features of NCERT Solutions for Class 12 Maths Chapter 4 Determinants
Class 12 NCERT Solutions of CoolGyan’S for Chapter 4 cover the following points and formulas.
- If any two rows or any two columns are identical or proportional, then the value of the determinant is zero
- A square matrix A has an inverse if and only if A is non-singular
- Unique solution of equation AX = B is given by X = A–1 B, where A ≠ 0
- If a system of equation is consistent, then it has a solution
- If a system of equations is inconsistent, then there is no solution
- For a square matrix A in matrix equation AX = B:
If | A| ≠ 0, then there exists a unique solution
If | A| = 0 and (adj A) B ≠ 0, then there exists no solution
If | A| = 0 and (adj A) B = 0, then system may or may not be consistent