Download the Matrices and Determinants Formulas below as a free PDF. The sheet contains every formula tested by CBSE in the 7-mark question block on Class 12 Mathematics Chapter 4 Determinants, plus the JEE Main extensions used in the same area. The Matrices and Determinants Formulas are structured for quick lookup.
- CBSE Weightage: 5 to 7 marks (one short answer on properties plus one MCQ on minors and cofactors).
- JEE Main Weightage: 3 to 5% of the Maths section (1 to 2 questions per paper).
- CUET (UG) Weightage: 1 to 2 MCQs in nearly every shift.
This Formula Sheet is curated by Class 12 Maths experts at Collegedunia, mapped to the 2026-27 NCERT edition, and refined against the last five years of CBSE Board and JEE Main papers.
How the Matrices and Determinants Formulas on the Matrices and Determinants Formulas Help You
The Matrices and Determinants Formulas address this in the same order as the NCERT textbook.
This sheet is built around the five habits CBSE toppers share when revising Chapter 4: expansion along the row with the most zeros, property-quoting in proofs, the cofactor sign rule, the adjoint identity, and Cramer's rule for non-homogeneous systems.
- 2026-27 NCERT Alignment: Every rule matches the current edition, covering order-2 and order-3 determinants, properties, area of a triangle, minors, cofactors, adjoint, inverse, and systems of linear equations.
- Sign-Rule Strip: The cofactor sign (-1)i+j sits at the top of the recall cards so a misplaced minus, the single most common 1-mark slip, is harder to make.
- Inverse Template: The three-step A-1 = 1|A| adj(A) appears with the existence condition |A| ≠ 0 , in the exact sequence CBSE awards the 5 marks.
- JEE Property Extensions: Rows for |kA| and |adj(A)| are kept on the same sheet so JEE aspirants do not need a separate handout.
Determinants Video Walkthrough
Source: Magnet Brains on YouTube
All Determinants Formulas for Class 12 Maths in One learn Table
The Matrices and Determinants Formulas address this in the same order as the NCERT textbook.
The table below catalogues every formula, property, and identity from Class 12 Maths Chapter 4, mapped to the exact NCERT section where it is introduced. Each row has surfaced in a CBSE or JEE Main question in the last five years, so anchor your revision around it.
| Concept | Formula / Rule | NCERT Section | Common Use |
|---|---|---|---|
| Determinant of order 2 | vmatrix a & b c & d vmatrix = ad - bc | 4.2 | 1-mark MCQ |
| Determinant of order 3 (row expansion) | |A| = a11 C11 + a12 C12 + a13 C13 | 4.2 | 3-mark CBSE |
| Determinant of order 3 (column expansion) | |A| = a11 C11 + a21 C21 + a31 C31 | 4.2 | 3-mark CBSE |
| Property 1: Transpose | |AT| = |A| | 4.3 | Property quote |
| Property 2: Row swap | Interchanging any two rows or columns flips the sign of |A| | 4.3 | Proof step |
| Property 3: Two identical rows | If any two rows or columns are identical, |A| = 0 | 4.3 | Assertion-reason |
| Property 4: Scalar multiple of a row | Multiplying one row by k multiplies |A| by k | 4.3 | Property quote |
| Property 5: Sum splitting | If a row is the sum of two row vectors, |A| splits as the sum of two determinants | 4.3 | Proof step |
| Property 6: Row operation invariance | Ri → Ri + k Rj leaves |A| unchanged (i ≠ j) | 4.3 | Simplification |
| Determinant of scalar multiple | |kA| = kn |A| for an n × n matrix | 4.3 (extension) | JEE Main |
| Determinant of product | |AB| = |A| · |B| | 4.3 (extension) | JEE Main |
| Determinant of inverse | |A-1| = 1|A| | 4.5 (extension) | JEE Main |
| Area of a triangle | Δ = 12 | vmatrix x1 & y1 & 1 x2 & y2 & 1 x3 & y3 & 1 vmatrix | | 4.3 | 2-mark CBSE |
| Collinearity test | Three points are collinear iff vmatrix x1 & y1 & 1 x2 & y2 & 1 x3 & y3 & 1 vmatrix = 0 | 4.3 | 2-mark CBSE |
| Minor Mij | Determinant of the sub-matrix after deleting row i and column j | 4.4 | Definition |
| Cofactor Cij | Cij = (-1)i+j Mij | 4.4 | 1-mark MCQ |
| Cofactor sign pattern (3x3) | pmatrix + & - & + - & + & - + & - & + pmatrix | 4.4 | Memory recall |
| Adjoint definition | adj(A) = [Cij]T (transpose of cofactor matrix) | 4.5 | 3-mark CBSE |
| Adjoint identity | A · adj(A) = adj(A) · A = |A| · In | 4.5 | 5-mark CBSE |
| Singular matrix | |A| = 0 : A-1 does not exist | 4.5 | Definition |
| Non-singular matrix | |A| ≠ 0 : A-1 exists | 4.5 | Existence check |
| Inverse via adjoint | A-1 = 1|A| adj(A) (when |A| ≠ 0) | 4.5 | 5-mark CBSE |
| Inverse of product | (AB)-1 = B-1 A-1 | 4.5 (extension) | JEE Main |
| Determinant of adjoint | |adj(A)| = |A|n-1 for an n × n matrix | 4.5 (extension) | JEE Main |
| Adjoint of adjoint | adj(adj(A)) = |A|n-2 A | 4.5 (extension) | JEE Main |
| Matrix form of a linear system | AX = B , with A the coefficient matrix, X the variable matrix, B the constants matrix | 4.6 | 5-mark CBSE |
| Unique solution (consistent) | |A| ≠ 0 : X = A-1 B | 4.6 | 5-mark CBSE |
| Cramer's rule (3 unknowns) | x = 1Δ, y = 2Δ, z = 3Δ, Δ ≠ 0 | 4.6 (extension) | JEE Main |
| No solution (inconsistent) | |A| = 0 and (adj A) B ≠ O | 4.6 | 5-mark CBSE |
| Infinitely many solutions | |A| = 0 and (adj A) B = O | 4.6 | 5-mark CBSE |
| Homogeneous system | AX = O : trivial solution when |A| ≠ 0; infinitely many non-trivial solutions when |A| = 0 | 4.6 | Assertion-reason |
Around 9 of these 38 rules cover the entire 5 to 7 mark CBSE budget for the Matrices and Determinants Formulas every year.
Determinants Quick-Fact Cards for Last-Minute Recall
The Matrices and Determinants Formulas address this in the same order as the NCERT textbook.
The seven cards below collapse the learn table into the highest-use atomic facts from Chapter 4. Memorise them as raw values; CBSE 1-mark MCQs and JEE Main extension questions almost always test one of these directly.
When to Use Which Determinants Formula in CBSE and JEE Numericals
Picking the wrong tool wastes the entire 5 marks of a long-answer question even when the arithmetic is right. The decision tree below maps the typical Class 12 Maths Chapter 4 question setup to the rule CBSE expects, with the existence check in front of every step.
- Evaluate a determinant: scan for a row or column with two zeros and expand along it. If no zeros are present, apply Property 6 ( Ri → Ri + k Rj ) to create them.
- Prove a determinant identity: use Property 5 (sum splitting) and Property 6 (row operations) and quote each property name as you apply it. The mark is for the property name, not the algebra.
- Find the area or test collinearity: use Δ = 12 | D | . Three points are collinear iff D = 0 . Always take the absolute value for area.
- Find the inverse: compute |A| first. If |A| = 0, write "inverse does not exist" and stop. Otherwise apply A-1 = 1|A| adj(A) .
- Solve a system by matrix method: write AX = B , compute |A|, then apply X = A-1 B when |A| ≠ 0, or analyse (adj A) B to decide between no-solution and infinite-solutions cases.
CBSE and JEE Numerical Pattern Templates for Determinants
The four templates below capture the question framings used most often in Class 12 Chapter 4. Match the setup of the paper question to a template, then jump straight to the formula on the learn table.
| Pattern | Setup | Use this rule |
|---|---|---|
| Inverse-and-solve | "Solve AX = B using A-1" | X = A-1 B , with A-1 = 1|A| adj(A) |
| Property-proof | "Prove |A| = 0 given a row identity" | Property 3 (identical rows) or Property 7 (proportional rows) |
| Area-of-triangle | "Find the area given three vertices" | Δ = 12 | D | |
| Adjoint extension (JEE) | "Find |adj A| given |A|" | |adj(A)| = |A|n-1 |
Memory Mnemonics for Class 12 Maths Determinants
Use these short-form mnemonics in the last hour before the exam. They map abstract properties to concrete actions, which is what CBSE marking schemes reward.
Top 5 Most-Asked Determinants Topics in Class 12 Board Exams
The ranking below shows which sub-topics of Determinants have driven the most marks in CBSE Class 12 Maths papers from 2025 back to 2021. Anchor your last revision pass on these five.
- Inverse of a 3x3 matrix via adjoint and solving AX = B (5 marks, almost every year).
- Property-based proof of a determinant identity (4 marks, 4 of last 5 years).
- Area of a triangle from three vertices using the determinant formula (2 marks, recurring).
- Cofactor and adjoint computation for a 3x3 matrix (3 marks, recurring).
- Singular vs non-singular identification by computing |A| (1 mark assertion-reason or MCQ).
Full year-wise PYQ map: Determinants Class 12 Maths NCERT Solutions
One-Shot Revision Tips for Class 12th Maths Chapter 4 Determinants
The five tips below summarise what to revise in the final 60 minutes before the board exam. Each one targets a specific high-frequency CBSE error pattern that costs Class 12 candidates marks every year.
- Always compute |A| first: begin every inverse or matrix-method answer with the determinant value. CBSE awards the first mark for the existence check.
- Expand along the smartest row or column: pick the one with the most zeros to kill product terms. This saves 30 to 60 seconds per question.
- Take absolute value for area: the area of a triangle is 12 | D | , not 12 D . A negative answer loses 1 mark.
- Cofactor matrix transposed gives adjoint: build [Cij] first, then transpose. The transpose step is worth 1 mark on the 3-mark adjoint question.
- Quote the property name in proofs: when applying Ri → Ri + k Rj, write the property number alongside the operation.
More Class 12 Maths Determinants Resources from Collegedunia
NCERT Formula Sheet for Class 12 Maths: All Chapters
Use the table below to jump to the Formula Sheet of any other Class 12 Maths chapter. Each link opens the same learn-table format you have on the Matrices and Determinants Formulas.
| Chapter | Formula Sheet |
|---|---|
| Chapter 1 | Relations and Functions Formula Sheet |
| Chapter 2 | Inverse Trigonometric Functions Formula Sheet |
| Chapter 3 | Matrices Formula Sheet |
| Chapter 5 | Continuity and Differentiability Formula Sheet |
| Chapter 6 | Application of Derivatives Formula Sheet |
| Chapter 7 | Integrals Formula Sheet |
| Chapter 8 | Application of Integrals Formula Sheet |
| Chapter 9 | Differential Equations Formula Sheet |
| Chapter 10 | Vector Algebra Formula Sheet |
| Chapter 11 | Three Dimensional Geometry Formula Sheet |
| Chapter 12 | Linear Programming Formula Sheet |
| Chapter 13 | Probability Formula Sheet |
Matrices and Determinants Formulas: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
Exercise-wise Breakdown of the Determinants Chapter
The Determinants chapter splits into 6 numbered exercises plus a Miscellaneous Exercise. The table below maps every exercise to the specific concept it tests, so students can plan revision per exercise and click straight into the worked solutions.
| Exercise | Topic Tested |
|---|---|
| Exercise 4.1 | Evaluation of 2x2 and 3x3 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 | Applications: solving systems of linear equations |
| Exercise 4.6 | Consistency of system of linear equations |
| Miscellaneous Exercise | Mixed determinant concepts and applications |
PDF Download Formats and Languages for the Determinants Chapter
The Determinants Class 12 PDF on this page is available in three formats - each suited to a different revision style. The table below summarises what each format is best for:
| Format | Best for | Approx. size |
|---|---|---|
| Normal-resolution PDF | Phone reading, quick revision between classes | 2-3 MB |
| HD PDF | Print-ready, desk study, board hall photocopy | 8-10 MB |
| Handwritten Notes PDF | Mirrors how a topper writes the chapter under Sunday-revision pace | 5-7 MB |
The determinants class 12 ncert pdf and the parallel Hindi-medium edition both follow the same notation and equation numbering as the printed NCERT 2026-27 release. Key points students should know:
- NCERT-faithful: Every definition, theorem and exercise on the determinants class 12 ncert pdf matches the printed textbook line for line.
- Hindi-medium edition: The determinants class 12 pdf is also available in Hindi - same page numbering, same equation labels.
- Formula PDF separate: The determinants class 12 formulas pdf is a one-page A4 reference sheet listing every identity used in the chapter.
- Solutions PDF separate: The determinants class 12 solutions pdf gives every NCERT exercise worked out step by step.
- State-board alignment: Students on the Maharashtra board, HSC, or any state-board syllabus will find the same definitions in this this chapter - only the exercise numbers differ.
Tip: Many toppers keep two parallel copies - a printed formula sheet on A4 for desk revision (the determinants class 12 formulas pdf), and the full these notes on a phone for commute revision. Both files are free and linked above.
Important Questions and Previous Year Trends for the Determinants Chapter
The most repeated question patterns in CBSE Class 12 Maths for the Determinants chapter have settled into a stable cluster across 2019 to 2024 boards. Three question templates account for over 80% of the marks this chapter contributes:
| Template | Typical Marks | What it tests |
|---|---|---|
| Proof / property verification | 3 marks | Students show that a given relation/function/expression satisfies the chapter's definitions. |
| One-step computation | 2 marks | Substitution-based item: plug into a known formula and simplify. |
| Case-study scenario | 4 marks | Real-world setup applying the chapter's definitions, introduced in CBSE 2021+ papers. |
Walking through one example of each template before the exam covers most of the predictable this Class 12 page important questions you will see on board day.
- the resource previous year questions for 2019-2024 are linked from the PYQ block at the bottom of this page - the exact CBSE phrasings.
- The the chapter notes important questions with solutions set is reused by toppers in the last fortnight of revision.
- For NCERT Exemplar practice, the matching determinants class 12 extra questions set adds advanced problems suitable for JEE Main and JEE Advanced.
- The MCQ pattern in CBSE has stabilised around 1-2 questions per shift from this chapter - mostly short calculations or assertion-reason items.
Year-wise PYQ Distribution
The table below maps the dominant question type asked from the Determinants chapter across recent CBSE Class 12 Maths boards:
| Year | Dominant Question Type | Approx. Marks |
|---|---|---|
| 2024 | Property verification + case-study item | 5-6 marks |
| 2023 | Computation with proof + assertion-reason MCQ | 5-6 marks |
| 2022 | Long-answer derivation + 2-mark substitution | 5-7 marks |
| 2021 | Definition recall + property check | 4-5 marks |
| 2020 | One-step computation + 3-mark proof | 5 marks |
The full the PDF important questions with solutions set (every year, every paper, every question type) is linked from the PYQ page at the bottom of this article.
How the Determinants Notes Pair with NCERT Solutions and the Formula Sheet
The this chapter notes work best when paired with two sister resources from the Class 12 Maths hub. The table below shows how each resource fits into a typical revision week:
| Resource | Use it for | When |
|---|---|---|
| Determinants Notes (this page) | Theory, definitions, exam patterns | First pass, before practice |
| determinants class 12 ncert solutions PDF | Step-by-step solved exercises | Second pass, during NCERT practice |
| determinants class 12 formulas PDF | One-page identity recall | Third pass, alongside mock papers |
| Handwritten Notes PDF | Quick reading in topper's handwriting | Anytime, especially commute revision |
Around 60 percent of the chapter's scoring vocabulary appears on all three pages, so cross-resource use reinforces recall without adding study time.
- The determinants class 12 ncert solutions cover every back-of-chapter exercise plus the miscellaneous exercise.
- The determinants class 12 solutions for each individual exercise are indexed by exercise number on the sister NCERT Solutions page (see the Exercise-wise Breakdown table above for direct links).
- The determinants class 12 formulas reference sheet is the same A4 file students sometimes refer to as these notes all formulas - it lists every identity used in the chapter.
- State-board references: RD Sharma, ML Aggarwal, Teachoo and the Maharashtra board this Class 12 page textbook PDF all share the same core definitions.
- For class-first search phrasings - class 12 determinants solutions, class 12 determinants ncert solutions, ncert class 12 determinants solutions - the same files cover the request.
Reference Books and State-Board Mapping
Students using reference books beyond NCERT, or studying under a state board, can map this chapter cleanly:
| Reference | How it maps to the resource |
|---|---|
| RD Sharma Class 12 Determinants | Question patterns overlap with NCERT at ~70%; an advanced supplement. |
| ML Aggarwal Class 12 Determinants | Solutions style is closer to JEE; good for problem-solving practice. |
| Teachoo the chapter notes | Free online walkthroughs; useful for video-style learning. |
| Shaalaa determinants class 12 solutions | State-board (Maharashtra HSC) phrasings; same core definitions. |
| Maharashtra board the PDF textbook PDF | Same chapter content under the HSC syllabus; exercise numbers differ. |
| NCERT Exemplar Class 12 Determinants | Advanced problems for JEE Main/JEE Advanced preparation. |
How to Use the Determinants Notes Page Most Effectively
The recommended study plan for the this chapter chapter splits across three sittings. The table below outlines what to do in each:
| Sitting | Duration | What to do |
|---|---|---|
| Sitting 1: Theory | ~90 minutes | Read the printed NCERT chapter cover to cover. Mark every definition and theorem statement. Then read the formula recall section on this page. |
| Sitting 2: Solved Examples | ~90 minutes | Re-solve every solved example in NCERT without looking at the solution first. Compare your steps against the printed working. Use the determinants class 12 ncert solutions PDF if stuck. |
| Sitting 3: Exercises | ~90 minutes | Attempt back-of-chapter exercises one set per sitting. Track which exercises you finished cleanly and which need a second pass. Click into the linked exercise pages above for verification. |
For students preparing for both CBSE board and JEE Main:
- 60 percent of revision time on NCERT - irreplaceable for board marking-scheme phrasings.
- 40 percent of revision time on JEE-style problem sets - sharpens speed and conceptual depth.
- The these notes important questions set on the previous-year page is the closest free analogue to a JEE-style problem set for this chapter.
- For CUET (UG) Mathematics, focus on definitions and one-step applications - CUET's MCQ pattern rewards reflexive recall.
Matrices and Determinants Formulas - Frequently Asked Questions
Ques. Where can I download the Matrices and Determinants Formulas for free?
Ans. You can download the Class 12 Maths Chapter 4 Determinants Formula Sheet PDF directly from the download card on this page. Both the Normal and HD versions are free and cover every formula listed in the learn table above, including the eight properties, the cofactor sign rule, the adjoint identity, and the Cramer's rule extension.
Ques. Is the Determinants formula sheet aligned with the 2026-27 NCERT?
Ans. Yes. Every formula matches the current 2026-27 syllabus for Class 12 Maths Chapter 4. The chapter retains determinants of order 2 and 3, properties, area of a triangle, minors, cofactors, adjoint, inverse, and the application to systems of linear equations. Extension rows for |kA|, |adj A|, and Cramer's rule are flagged so JEE aspirants get the full coverage.
Ques. How many pages is the Class 12 Maths Chapter 4 Determinants Formula Sheet PDF?
Ans. The Determinants Formula Sheet PDF runs 7 pages and packs every expansion rule, property, cofactor and adjoint identity, inverse formula, area-of-triangle shortcut, and Cramer's-rule template from Chapter 4, plus the quick-fact card grid and the property-recall strip.
Ques. What is the formula for the inverse of a matrix using determinants?
Ans. The inverse formula is A-1 = 1|A| adj(A) , valid only when |A| ≠ 0 (the matrix is non-singular). Use it in three steps: compute |A|, build the cofactor matrix and transpose it to get adj(A), then divide by |A|. CBSE marking schemes award 1 mark for each step plus 2 marks for accurate arithmetic.
Ques. What is the cofactor sign rule for a 3x3 determinant?
Ans. The cofactor Cij = (-1)i+j Mij , where Mij is the minor.
For a 3x3 matrix, the sign pattern is a checkerboard pmatrix + & - & + - & + & - + & - & + pmatrix , starting with + at position (1, 1). Misplacing one sign is the most common 1-mark MCQ slip in Class 12 Maths Chapter 4.
Ques. How do I find the area of a triangle using the determinants formula?
Ans. Given vertices (x1, y1), (x2, y2), (x3, y3), the area is Δ = 12 | vmatrix x1 & y1 & 1 x2 & y2 & 1 x3 & y3 & 1 vmatrix | . Always take the absolute value, a negative answer loses 1 mark. The three points are collinear iff the determinant evaluates to 0.
Ques. What is the determinant of the adjoint of a matrix?
Ans. For an n × n matrix A, |adj(A)| = |A|n-1 . For a 3x3 matrix this gives |adj(A)| = |A|2 , and for a 2x2 matrix it gives |adj(A)| = |A| . This identity is a recurring JEE Main 1-mark MCQ and a CBSE assertion-reason favourite.
Ques. When does a system of linear equations have a unique solution by the matrix method?
Ans. Writing the system as AX = B, there is a unique solution iff |A| ≠ 0 (the coefficient matrix is non-singular), and the solution is X = A-1 B. If |A| = 0 and (adj A) B = O, the system has infinitely many solutions; if |A| = 0 and (adj A) B ≠ O, the system is inconsistent.
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