NCERT Solutions Class 12 Maths Chapter 4 provides detailed solutions to all 61 questions on Determinants, covering properties, cofactors, adjoint matrices, and applications in solving linear equations. You’ll learn how to calculate determinants of different orders, apply properties to simplify complex problems, find inverse matrices using adjoint method, and solve systems of equations using Cramer’s rule. These skills are essential for competitive exams and form the foundation for linear algebra in engineering and sciences.
Download Complete Chapter 4 Solutions PDF
All exercises with step-by-step solutions | Updated 2025-26 | Free Download
Download PDF (Free)NCERT Solutions Class 12 Maths Chapter 4 Determinants – Complete Guide
NCERT Class 12 Chapter 4 on Determinants introduces you to one of the most powerful tools in linear algebra that forms the foundation for advanced mathematical concepts. You’ll explore how determinants are numerical values associated with square matrices and learn systematic methods to calculate them for 2×2 and 3×3 matrices using expansion techniques. This chapter carries 5 marks weightage in CBSE board exams, typically featuring questions on properties of determinants, area of triangles using determinants, and solving linear equations.
📊 CBSE Class 12 Maths Chapter 4 – Exam Weightage & Marking Scheme
| CBSE Board Marks | 5 Marks |
| Unit Name | Algebra |
| Difficulty Level | Medium |
| Importance | Medium |
| Exam Types | CBSE Board, State Boards |
| Typical Questions | 1-2 questions |
You will discover the fascinating properties of determinants that make calculations easier, such as how row operations affect determinant values and the relationship between determinants of transpose and original matrices. The chapter thoroughly covers minors, cofactors, and adjoint of matrices, which are essential for finding inverse matrices. You’ll learn to apply these concepts to solve systems of linear equations using Cramer’s rule, a method particularly useful when dealing with two or three variables.
The practical applications of determinants extend beyond pure mathematics into physics, engineering, and economics, where they help analyze linear transformations, calculate areas and volumes, and solve optimization problems. CBSE exams commonly include 2-mark questions on properties, 3-mark problems on solving equations, and occasionally 5-mark questions combining multiple concepts.
Quick Facts – Class 12 Chapter 4
| 📖 Chapter Number | Chapter 4 |
| 📚 Chapter Name | Determinants |
| ✏️ Total Exercises | 6 Exercises |
| ❓ Total Questions | 61 Questions |
| 📅 Updated For | CBSE Session 2025-26 |
Mastering this chapter will strengthen your problem-solving abilities and prepare you for higher mathematics in competitive exams like JEE and engineering entrance tests. With consistent practice of NCERT solutions and previous year questions, you’ll develop the confidence to tackle any determinant-related problem efficiently and accurately in your board examinations.
NCERT Solutions Class 12 Maths Chapter 4 – All Exercises PDF Download
Download exercise-wise NCERT Solutions PDFs for offline study
| Exercise No. | Topics Covered | Download PDF |
|---|---|---|
| Exercise 4.1 | Complete step-by-step solutions for 8 questions | 📥 Download PDF |
| Exercise 4.2 | Complete step-by-step solutions for 5 questions | 📥 Download PDF |
| Exercise 4.3 | Complete step-by-step solutions for 5 questions | 📥 Download PDF |
| Exercise 4.4 | Complete step-by-step solutions for 18 questions | 📥 Download PDF |
| Exercise 4.5 | Complete step-by-step solutions for 16 questions | 📥 Download PDF |
| Miscellaneous Exercise on Chapter 4 | Complete step-by-step solutions for 9 questions | 📥 Download PDF |
Determinants – Key Formulas & Concepts
Quick reference for CBSE exams
| Formula | Description | When to Use |
|---|---|---|
| Determinant of a 2×2 Matrix \\[\\begin{vmatrix} a & b \\\\ c & d \\end{vmatrix} = ad – bc\\] | Calculates the determinant of a 2×2 matrix. Note: Multiply the diagonal elements and subtract the product of the off-diagonal elements. Remember the order of subtraction: (a*d) – (b*c) | Finding the area of a triangle given its vertices, checking if a matrix is invertible (if determinant is non-zero), solving systems of linear equations (Cramer’s Rule) |
| Determinant of a 3×3 Matrix \\[\\begin{vmatrix} a_1 & b_1 & c_1 \\\\ a_2 & b_2 & c_2 \\\\ a_3 & b_3 & c_3 \\end{vmatrix} = a_1\\begin{vmatrix} b_2 & c_2 \\\\ b_3 & c_3 \\end{vmatrix} – b_1\\begin{vmatrix} a_2 & c_2 \\\\ a_3 & c_3 \\end{vmatrix} + c_1\\begin{vmatrix} a_2 & b_2 \\\\ a_3 & b_3 \\end{vmatrix}\\] | Calculates the determinant of a 3×3 matrix by expanding along the first row. Note: Remember the alternating signs: + – +. You can expand along any row or column, but the signs must alternate correctly. Choose the row or column with the most zeros to simplify calculation. | Finding the volume of a parallelepiped, checking linear independence of three vectors, solving systems of three linear equations. |
| Area of a Triangle (using Determinants) \\[Area = \\frac{1}{2} \\begin{vmatrix} x_1 & y_1 & 1 \\\\ x_2 & y_2 & 1 \\\\ x_3 & y_3 & 1 \\end{vmatrix}\\] | Calculates the area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3). Note: The area is always positive. If the determinant is negative, take the absolute value. If the determinant is zero, the points are collinear. | Given the coordinates of the vertices of a triangle, find its area. Useful when the triangle is not right-angled. |
| Condition for Collinearity \\[\\begin{vmatrix} x_1 & y_1 & 1 \\\\ x_2 & y_2 & 1 \\\\ x_3 & y_3 & 1 \\end{vmatrix} = 0\\] | Checks if three points (x1, y1), (x2, y2), and (x3, y3) are collinear (lie on the same line). Note: Calculate the determinant. If it’s zero, the points are collinear. | Proving that three points lie on the same line. |
| Adjoint of a Matrix (adj A) \\[adj(A) = (Cofactor(A))^T\\] | The adjoint of a matrix is the transpose of its cofactor matrix. Note: First find the matrix of cofactors. Then, take the transpose (swap rows and columns). Remember to find the cofactors with the correct signs. | Finding the inverse of a matrix, solving systems of linear equations. |
| Inverse of a Matrix (A⁻¹) \\[A^{-1} = \\frac{1}{|A|} adj(A)\\] | Calculates the inverse of a square matrix A. Note: A matrix is invertible if and only if its determinant is non-zero (|A| ≠ 0). Divide each element of the adjoint matrix by the determinant of A. | Solving matrix equations, finding the solution to a system of linear equations represented in matrix form (AX = B => X = A⁻¹B). |
| Solving Linear Equations (Matrix Method) \\[AX = B \\implies X = A^{-1}B\\] | Solves a system of linear equations represented in matrix form, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix. Note: First, write the system of equations in matrix form. Then, find the inverse of the coefficient matrix (A⁻¹). Finally, multiply A⁻¹ by B to find X. | Solving systems of linear equations with the same number of equations as variables. |
| Properties of Determinants (k * A) \\[|kA| = k^n |A|\\] | If a matrix A of order n is multiplied by a constant k, then the determinant of the new matrix is k^n times the determinant of A. Note: n is the order of the matrix (e.g., 2 for a 2×2 matrix, 3 for a 3×3 matrix). k is raised to the power of the order of the matrix. | Simplifying determinant calculations when a common factor can be taken out of a row or column. |
| Properties of Determinants (Product of Matrices) \\[|AB| = |A| |B|\\] | The determinant of the product of two matrices is equal to the product of their determinants. Note: This property is useful for problems where you need to find the determinant of a product of matrices without actually performing the matrix multiplication. | Simplifying determinant calculations when dealing with matrix multiplication. |
| Determinant of Transpose \\[|A^T| = |A|\\] | The determinant of the transpose of a matrix is equal to the determinant of the original matrix. Note: Taking the transpose does not change the value of the determinant. | Simplifying calculations involving transpose of a matrix. |
Frequently Asked Questions – NCERT Class 12 Maths Chapter 4
📚 Related Study Materials – Class 12 Maths Resources
| Resource | Access |
|---|---|
| NCERT Class 12 Sociology Textbook | Download Book |
| NCERT Class 10 Maths (Foundation) | View Solutions |