CBSE Class 12 Maths Chapter 4 Determinants – NCERT Solutions 2025–26

• Exercise 4.1: This exercise introduces the concept of determinants and their notation. Students will learn how to find the determinants of 2x2 and 3x3 matrices using various methods, including the rule of Sarrus and the cofactor expansion method.

• Exercise 4.2: This exercise focuses on the application of determinants in solving systems of linear equations. Students will learn about Cramer's rule, which allows them to solve systems of linear equations using determinants.

• Exercise 4.3: In this exercise, students will learn about the inverse of a matrix and how to find it using determinants. They will also practice using determinants to find the area of a triangle and the volume of a parallelepiped.

• Exercise 4.4: This exercise covers the concept of minors and cofactors of a matrix. Students will learn how to find the minors and cofactors of a matrix and use them to find the inverse of a matrix.

• Exercise 4.5: In this exercise, students will learn about the adjoint of a matrix and its properties. They will also practice finding the adjoint of a matrix and using it to find the inverse of a matrix.

NCERT Solutions for Class 12 Maths – Free PDF Download

Chapter 4 – Determinants

4.1 Introduction

In Chapter 4 of Class 12 Maths NCERT Solutions, you will learn about determinants and their basic concepts and theorems. The chapter covers determinants up to order three with real entries, as well as various properties of determinants such as minors, cofactors, and applications of determinants. Additionally, the chapter includes topics like the inconsistency of the system of linear equations and solutions of linear equations in two or three variables using the inverse of a matrix.

4.2 Determinant

In Chapter 4 of Class 12 Maths, students can learn how to find the determinant of a matrix. This includes understanding the determinant of a matrix of order one, which is relatively easy to comprehend. The section also covers the determination of the matrix of order two and order three. It explains how to expand the determinant along the row or column containing the maximum number of zeros. Additionally, it helps in understanding positive multiplication rather than negative multiplication.

4.3 Area of Triangle

The Determinants Class 12 PDF covers how to calculate the area of a triangle using vertices and expressions. It explains the concept of the area as a positive quantity and the absolute value of the determinant. It also discusses the use of positive and negative values of the determinant in calculations.

4.4 Minors and Cofactors

This section teaches students how to simplify the expansion of a determinant using minors and cofactors. It also explains the definitions of minors and cofactors. The text covers five methods to calculate the area of a triangle, including using the sum of the product of elements of any row or column with their corresponding cofactor.

4.5 The Adjoint and Inverse of a Matrix

This section is about matrices, focusing on their inverse and existence. It explains how to find the adjoint of a matrix and provides five definitions, including the adjoint of a square matrix and the theorem to verify it. Students will learn how to discover and apply these theorems to verify the definitions of various types of problems.

4.6 Applications of Determinants and Matrices

In Chapter 4 maths class 12, students learn about using determinants and matrices to solve systems of linear equations with two or three variables. The chapter covers checking if the system of equations is consistent, explaining both consistent and inconsistent systems in detail. Students also learn how to solve these equations using the inverse of a matrix, express linear equations as matrix equations, and solve them using the inverse of the coefficient matrix. The chapter also discusses the uniqueness of matrix equations in providing unique solutions for systems of equations, known as the matrix method.

The NCERT Solutions for Class 12 Maths Chapter 4 Determinants' Key features

The following concepts and equations are covered in the NCERT Solutions for chapter 4 Maths class 12 at Vedantu.

• The determinant has zero value if any two rows or any two columns are proportionate or identical.

• If and only if A is not singular, a square matrix A has an inverse.

• When A ≠ 0, the unique solution to the equation AX = B is X = A–1 B.

• A system of equations has a solution if it is consistent.

• There cannot be a solution to an inconsistent set of equations.

• In the matrix equation AX = B, with a square matrix A:

There is a unique solution if | A| ≠ 0.

If (adj A) B ≠ 0 and |A| = 0, then no solution exists.

In case (adj A) B = 0 and | A| = 0, the consistency of the system is uncertain.

Overview of Deleted Syllabus for CBSE Class 12 Determinants 2024-25

Class 12 Maths Chapter 4: Exercises Breakdown

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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.

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