These NCERT notes for Determinants Class 12 Maths pack all of Chapter 4 into one exam-ready summary: expansion, properties, minors, cofactors, adjoint, inverse and systems of equations. You can revise the whole chapter in under an hour. The free notes PDF is available on this page.
The notes below follow the NCERT rationalised syllabus and CBSE guidelines for 2026-27.
- CBSE Boards: 6 to 8 marks alone, and 10 to 13 marks paired with Matrices in the Algebra unit.
- JEE Main: Matrices and Determinants together carry 2 to 3 questions per shift, about 6 to 8% of the paper.
- CUET (UG) Maths: 2 to 3 direct MCQs on properties, adjoint, inverse and Cramer's rule each cycle.
Every line of these Determinants Class 12 Notes is prepared by Collegedunia subject experts, mapped to the 2026-27 NCERT syllabus, and checked against the last five CBSE marking schemes.
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Table of Contents |
How the Determinants Class 12 Notes Help You Revise for Boards
CBSE has asked an adjoint-and-inverse question in 4 of the last 5 board papers. These notes give you visual recall of every property and a fixed method for adjoint, inverse and linear systems.
- The seven properties P1 to P7 are stated in one line each for single-page recall.
- Adjoint and inverse are laid out as a 5-step answer, the format CBSE rewards.
- The matrix method X = A-1B and Cramer's rule are contrasted so you pick the faster route.
Determinants Class 12 Video Walkthrough
Source: Magnet Brains on YouTube
Determinants Class 12 Most Important Topics for Board, JEE and CUET
The chapter moves from the determinant itself, to properties, to inverse algebra, to linear systems. Learn these core ideas first.
- Determinant: for a 2×2, |A| = ad - bc; for a 3×3, expand along any row using cofactors.
- Minor and cofactor: the minor Mij deletes row i and column j; the cofactor is Cij = (-1)i+j Mij.
- Adjoint and inverse: adj(A) is the transpose of the cofactor matrix, and A-1 = 1|A| adj(A) when |A| ≠ 0.
- Systems of equations: solve AX = B by the matrix method or Cramer's rule.
| The Seven Properties of Determinants | |
|---|---|
| P1: |AT| = |A| | P2: swapping two rows flips the sign |
| P3: two equal rows give |A| = 0 | P4: |kA| = kn|A| |
| P5: a row of sums splits the determinant | P6: Ri → Ri + k Rj keeps it unchanged |
| P7: a row or column of zeros gives |A| = 0 | |
Determinants Class 12 Formula Recall Cheat Sheet
| # | Formula | When to use |
|---|---|---|
| 1 | |A| = a11C11 + a12C12 + a13C13 | 3×3 cofactor expansion |
| 2 | |kA| = kn|A| | Scaling an n×n matrix |
| 3 | A · adj(A) = |A| In | Inverse and |adj A| = |A|n-1 |
| 4 | A-1 = 1|A| adj(A) | 4-mark inverse by adjoint |
| 5 | xi = |Ai||A| | Cramer's rule, one variable |
Determinants Class 12 Topic-wise CBSE Weightage
| Sub-topic | CBSE intensity |
|---|---|
| Expansion, minors, cofactors | Medium (1 to 2 mark MCQ) |
| Seven properties (especially P6) | High (3 to 4 mark proof) |
| Area of triangle, collinearity | Medium (2 mark direct) |
| Adjoint and inverse via adjoint | High (4 mark templated answer) |
| Matrix method for AX = B | High (5 mark long-answer) |
| Cramer's rule, consistency | Medium (3 to 4 mark sub-part) |
Determinants Class 12 Most Repeated Board Exam Questions
- If order 3 and |A| = 4, find |adj(A)|. Use |adj(A)| = |A|n-1 = 42 = 16. (CBSE 2024, 2022, 2019)
- Solve a 3-variable system by the matrix method. Write AX = B, find adj(A), then X = A-1B. (CBSE 2024, 2023, 2020)
- Prove a determinant identity using properties. Apply R1 → R1 + R2 + R3 to pull out a common factor. (CBSE 2023, 2021, 2018)
- Find k for a non-trivial solution of a homogeneous system: set |A| = 0 and solve. (CBSE 2022, 2019)
- Check collinearity of three points using the area determinant; it evaluates to zero. (CBSE 2021, 2020)
Determinants Class 12: Common Misconceptions Students Mix Up
| Wrong idea | The correct rule |
|---|---|
| Adjoint equals the cofactor matrix | The adjoint is the transpose of the cofactor matrix. |
| |kA| = k|A| | For an n×n matrix, |kA| = kn|A|. |
| A homogeneous system has a unique solution | AX = O has a non-trivial solution only if |A| = 0. |
| Row op Ri → Ri + k Rj changes the value | By P6 the determinant stays unchanged. |
Other Resources for Determinants Class 12
| Chapter | Notes | NCERT Solutions |
|---|---|---|
| Ch 1 Relations and Functions | Relations and Functions Notes | Relations and Functions Solutions |
| Ch 2 Inverse Trigonometric Functions | Inverse Trigonometry Notes | Inverse Trigonometry Solutions |
| Ch 3 Matrices | Matrices Notes | Matrices Solutions |
| Ch 4 Determinants | Determinants Notes | Determinants Solutions |
| Ch 5 Continuity and Differentiability | Continuity Notes | Continuity Solutions |
Exercise-wise Breakdown of the Determinants Chapter
| Exercise | Topic Tested |
|---|---|
| Exercise 4.1 | Evaluation of 2×2 and 3×3 determinants |
| Exercise 4.2 | Properties of determinants; area of a triangle |
| Exercise 4.3 | Minors and cofactors |
| Exercise 4.4 | Adjoint and inverse of a matrix |
| Exercise 4.5 | Solving systems of linear equations |
| Miscellaneous Exercise | Mixed determinant concepts and applications |
PDF Download Formats for the Determinants Notes
| Format | Best for | Approx. size |
|---|---|---|
| Normal-resolution PDF | Phone reading and quick revision | 2-3 MB |
| HD PDF | Print-ready desk study | 8-10 MB |
| Handwritten Notes PDF | Reading in a topper's handwriting | 5-7 MB |
Every definition and formula in the PDF matches the printed NCERT textbook. A Hindi-medium edition and a separate one-page formula sheet are also linked on the site.
How the Determinants Notes Pair with Solutions and the Formula Sheet
| Resource | Use it for | When |
|---|---|---|
| Notes (this page) | Theory, definitions, exam patterns | First pass, before practice |
| NCERT Solutions PDF | Step-by-step solved exercises | Second pass, during practice |
| Formula Sheet PDF | One-page identity recall | Third pass, with mock papers |
How to Use the Determinants Notes Page Effectively
- Sitting 1 (theory): read the NCERT chapter, mark every property, then read the formula table above.
- Sitting 2 (examples): re-solve every solved example without looking, then check against the solutions PDF.
- Sitting 3 (exercises): attempt one exercise set per sitting and use the linked exercise pages to verify.
- Split revision roughly 60% NCERT and 40% JEE-style problems; for CUET focus on definitions and one-step MCQs.
Student Feedback - Class 12 Determinants Difficulty (Collegedunia Survey, March 2026):
- 68% of 900 students surveyed rated the 5-mark matrix-method system as the toughest question in Chapter 4.
- Across 1,100 answer scripts, students lost an average of 1.6 marks by skipping the transpose step in the adjoint.
- 71% of JEE aspirants said the |kA| = kn|A| rule is the property they re-check most.
Determinants Class 12 Notes - Frequently Asked Questions
What is the weightage of Determinants in CBSE Class 12 Maths Board Exam 2026?
Determinants alone carries about 6 to 8 marks. With Matrices, the Algebra unit contributes 10 to 13 marks, usually one MCQ, a 3-mark property proof and a 4 or 5-mark adjoint, inverse or matrix-method question.
What are the seven properties of determinants?
Transpose keeps the value; swapping two rows flips the sign; equal rows give zero; a scalar k on one row multiplies the value by k; a row of sums splits the determinant; adding a multiple of one row to another leaves it unchanged; a row of zeros gives zero.
How do you find the inverse of a matrix using its adjoint?
Compute the determinant and confirm it is non-zero. Find every cofactor, transpose the cofactor matrix to get the adjoint, then divide by the determinant: A inverse equals adj(A) divided by |A|.
When is a system of linear equations inconsistent?
For AX = B, if |A| = 0 and (adj A)B is non-zero, the system has no solution and is inconsistent. If |A| = 0 and (adj A)B is zero, it has infinitely many solutions. A non-zero |A| gives a unique solution.
Should I study Determinants before or after Matrices?
Always after Matrices. Chapter 3 introduces matrix multiplication and the transpose, which Chapter 4 uses inside the properties and the adjoint identity. Studying them in NCERT order matches how CBSE frames the combined questions.
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