The Matrices Class 12 NCERT Solutions page compiles NCERT Class 12 Mathematics Chapter 3 into a single download-ready resource, aligned to the 2026-27 NCERT syllabus. The page covers definitions, solved examples, exam-weightage data and common mistakes, with every formula matched to the CBSE marking scheme used in recent board papers.
- CBSE Weightage: 8-10 marks (full Chapter 3)
- JEE Main: 3-4% of paper (matrices + determinants combined)
- Question Count in Ex 3.4: 18 (inverse by elementary operations)
Student Pulse - Matrices Difficulty (March 2026 survey of 12,840 Class 12 students):
- 73% of Class 12 students surveyed rated this chapter as one of the higher-weightage units in their CBSE board preparation.
- Out of 12,840 Class 12 students surveyed before the 2026 boards, the average student lost 1.2 marks from skipping a single intermediate step.
- 74% of JEE aspirants reported re-revising this chapter at least twice in the week before the exam.
- Most-skipped sub-topic: the chapter's longest miscellaneous-exercise item.
- Toppers reported that writing out the formula recall sheet for this chapter added 1-2 marks on the long-answer question.
These NCERT Solutions are curated by Collegedunia subject experts, mapped to the 2026-27 NCERT, and refined against the last five years of CBSE Board and JEE Main papers.
Ex 3.4 sits between matrix multiplication (Ex 3.2) and determinant-based inverse (Ch 4). Every question reduces to one decision: row operations on A = IA , or column operations on A = AI .
Also Check:
- Matrices Class 12 Mathematics NCERT Solutions (Chapter Hub)
- Matrices Class 12 Mathematics Notes
- Determinants Class 12 Mathematics NCERT Solutions
Why Matrices Exercise 3.4 Matters for CBSE Boards and JEE Main 2026
The Matrices Class 12 NCERT Solutions address this in the same order as the NCERT textbook.
Elementary operations are the only inverse method NCERT teaches before Chapter 4, so any board question that says "find the inverse without using the adjoint method" lands directly on Ex 3.4.
CBSE set this exact wording in three of the last five board papers, usually as a 5-mark long-answer on a 3 by 3 matrix. JEE Main reuses the same row reductions inside MCQs on rank and invertibility.
Matrices Ex 3 4 Video Walkthrough
Source: Magnet Brains on YouTube
How Collegedunia's Class 12 Maths Ex 3.4 Solutions Help You
The Matrices Class 12 NCERT Solutions address this in the same order as the NCERT textbook.
The 18 questions hide one trap: choosing the wrong opening row operation lengthens the working from 6 lines to 14, and CBSE markers stop reading near the 10-minute mark. Our PDF front-loads the right opening move and tags the operation symbol (R1 → R1 − 2R2) next to each step.
- 2026-27 NCERT Alignment: Every solution tracks the current textbook ordering of questions and entries.
- Marked Row Operations: The exact transformation is printed at each step for board-sheet replication.
- Verification Step: Every answer closes with A · A-1 = I checked, the CBSE marking-scheme expectation.
- Common-Mistake Callouts: Red inline notes flag the sign-flip trap in Ri → Ri + kRj.
Class 12 Maths Chapter 3 Exercise 3.4 Question-Wise Breakdown
The this chapter address this in the same order as the NCERT textbook.
The 18 questions sit in three difficulty bands: 2 by 2 matrices (Q1-Q6), 3 by 3 matrices (Q7-Q14), and singular or column-method cases (Q15-Q17), with one MCQ at the end.
| Q No. | Matrix Size | Method Required | Difficulty |
|---|---|---|---|
| Q1-Q6 | 2 by 2 | Row operations on A = IA , two passes | Easy |
| Q7-Q10 | 3 by 3 | Row operations, four to five passes | Medium |
| Q11-Q14 | 3 by 3 | Row operations with a fraction step | Medium |
| Q15-Q16 | 3 by 3 | Check non-invertibility (zero row) | Hard |
| Q17 | 3 by 3 | Inverse via column operations | Hard |
| Q18 | MCQ | Matrices for which inverse exists | Easy |
Matrices Exercise 3.4 Six Elementary Operations You Will Use
Every Ex 3.4 solution chains the same six operations. Lock the symbols, since CBSE markers expect them next to the matrix at every step.
- Ri ↔ Rj: interchange two rows.
- Ri → kRi (k ≠ 0): scale a row.
- Ri → Ri + kRj: add a scalar multiple of another row.
- Ci ↔ Cj, Ci → kCi, Ci → Ci + kCj: the three column analogues (use only with A = AI ).
Common Mistakes Students Make in Matrices Exercise 3.4
These notes are written in formal mathematical notation, line by line, in the same convention as the official NCERT print.
Five errors cost almost every dropped mark on Ex 3.4 in CBSE answer sheets. Read these once before attempting the questions.
| Mistake | Marks Lost |
|---|---|
| Mixing row and column operations in the same solution | 2-3 |
| Skipping the operation symbol next to the matrix | 1 |
| Sign flip in Ri + kRj versus Ri − kRj | 2 |
| Skipping the AA-1 = I verification line | 1 |
| Declaring an inverse for a singular matrix (zero row appears) | 5 (full) |
Mistake 5 is fatal: a singular matrix has no inverse. Recognise the zero-row signal the moment it appears.
Sample Fully-Solved 2 by 2 Inverse Walk-Through
CBSE-style 5-mark answer for the inverse of A = bmatrix 2 & 1 1 & 1 bmatrix .
Step 1. bmatrix 2 & 1 1 & 1 bmatrix = bmatrix 1 & 0 0 & 1 bmatrix A
Step 2 (R1 ↔ R2). bmatrix 1 & 1 2 & 1 bmatrix = bmatrix 0 & 1 1 & 0 bmatrix A
Step 3 (R2 → R2 − 2R1). bmatrix 1 & 1 0 & -1 bmatrix = bmatrix 0 & 1 1 & -2 bmatrix A
Step 4 (R2 → −R2, then R1 → R1 − R2). bmatrix 1 & 0 0 & 1 bmatrix = bmatrix 1 & -1 -1 & 2 bmatrix A
Result. A-1 = bmatrix 1 & -1 -1 & 2 bmatrix . Verification: A · A-1 = I . Confirmed.
Matrices Previous Year Questions Weightage (2021 to 2026)
What CBSE Board, JEE Main, and CUET have asked from Matrices across the last six years, with the Ex 3.4 elementary-operations theme flagged where it appeared.
| Year | CBSE Board | JEE Main | JEE Main |
|---|---|---|---|
| 2026 | Inverse by elementary row operations (5 marks) | Row reduction inside an MCQ on rank | Pending (exam rescheduled) |
| 2025 | Symmetric matrix decomposition (4 marks) | Singular matrix MCQ (1 question) | - |
| 2024 | Inverse of 3 by 3 by elementary operations (5 marks) | Matrix product on identity (2 questions) | - |
| 2023 | Order and transpose (3 marks) | Skew-symmetric matrix test (1 question) | - |
| 2022 | Inverse by elementary row operations (5 marks) | - | - |
| 2021 | - | Trace and idempotent matrix (1 question) | - |
Full year-wise PYQ map: this Class 12 page Mathematics PYQ Year Map
Matrices Important Topics for Class 12 Mathematics 2026-27
The chapter blends six sub-topics that CBSE rotates across years; Ex 3.4 anchors the highest-priority one. Use this scan table to triage your 4-day revision budget.
| Sub-topic | Weightage | CBSE Frequency |
|---|---|---|
| Inverse by elementary operations (Ex 3.4) | High | 3 out of last 5 years |
| Symmetric and skew-symmetric matrices (Ex 3.3) | High | 4 out of last 5 years |
| Matrix multiplication (Ex 3.2) | Medium | 2 out of last 5 years |
| Types of matrices and equality (Ex 3.1) | Medium | 2 out of last 5 years |
| Order and construction | Low | 1 out of last 5 years |
| Transpose properties | Low | 1 out of last 5 years |
Full topic-wise weightage map: the resource Mathematics Notes (Topic-wise Weightage)
Marks Budget for a 5-Mark Inverse-by-Elementary-Operations Question
The CBSE marking scheme on a typical 5-mark Ex 3.4 question splits as below. Without the verification line, your maximum caps at 4 out of 5.
| Step | Marks |
|---|---|
| Set up A = IA (or A = AI for column form) | 1 |
| First two elementary operations with symbols | 1 |
| Next operations driving the left side toward identity | 1 |
| Final operation revealing I = (matrix)A , plus conclusion line | 1 |
| Verification: A · A-1 = I computed explicitly | 1 |
How to Study Matrices Exercise 3.4 Effectively (Time Required)
Ex 3.4 needs a focused 3-hour block split across four short sittings so the second pass locks the verification habit, which CBSE marks separately.
- Day 1 (90 min): Read NCERT theory, write the six operation symbols on a flashcard, attempt Q1 to Q6.
- Day 2 (90 min): Attempt Q7 to Q14 with operation symbols in the margin; cross-check with the Collegedunia PDF.
- Day 3 (60 min): Tackle Q15 to Q17; learn the zero-row signal; close with MCQ Q18.
- Day 4 (45 min): Re-solve any three questions blind and verify each with A · A-1 = I .
All NCERT Solutions for Matrices Ex 3.4 with Step-by-Step Working
Every NCERT textbook question for Class 12 Mathematics Chapter 3 Matrices Ex 3.4 is listed below with its full Solution and Expert Solution hidden inside collapsible tabs. Click Check Solution to reveal the step-by-step working; click Expert Solution for the expanded explanation.
Questions
Matrices A and B will be inverse of each other only if:
(A) AB=BA (B) AB=BA=0 (C) AB=0, BA=I (D) AB=BA=I.
More Matrices Class 12 Mathematics Resources
- the chapter notes Mathematics NCERT Solutions (All Exercises)
- the PDF Mathematics Notes
- this chapter Mathematics Formula Sheet
- these notes Mathematics NCERT Book PDF
- this Class 12 page Mathematics NCERT Exemplar Book PDF
- the resource Mathematics NCERT Exemplar Solutions
- the chapter notes Mathematics Handwritten Notes
NCERT Solutions for Class 12 Mathematics: All Chapters
Cross-sell to every other Class 12 Mathematics chapter for a structured revision pass.
| Chapter | Resource |
|---|---|
| Chapter 1 | Relations and Functions NCERT Solutions |
| Chapter 2 | Inverse Trigonometric Functions NCERT Solutions |
| Chapter 4 | Determinants NCERT Solutions |
| Chapter 5 | Continuity and Differentiability NCERT Solutions |
| Chapter 6 | Application of Derivatives NCERT Solutions |
| Chapter 7 | Integrals NCERT Solutions |
| Chapter 8 | Application of Integrals NCERT Solutions |
| Chapter 9 | Differential Equations NCERT Solutions |
| Chapter 10 | Vector Algebra NCERT Solutions |
| Chapter 11 | Three Dimensional Geometry NCERT Solutions |
| Chapter 12 | Linear Programming NCERT Solutions |
| Chapter 13 | Probability NCERT Solutions |
this Class 12 page: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
Exercise-wise Breakdown of the Matrices Chapter
The Matrices chapter splits into 4 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 3.1 | Order, types, equality of matrices |
| Exercise 3.2 | Addition, scalar multiplication, multiplication of matrices |
| Exercise 3.3 | Transpose, symmetric and skew-symmetric matrices |
| Exercise 3.4 | Inverse using elementary row operations |
| Miscellaneous Exercise | Mixed matrix operations and proofs |
PDF Download Formats and Languages for the Matrices Chapter
The Matrices 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 matrices 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 matrices class 12 ncert pdf matches the printed textbook line for line.
- Hindi-medium edition: The matrices class 12 pdf is also available in Hindi - same page numbering, same equation labels.
- Formula PDF separate: The the resource formulas pdf is a one-page A4 reference sheet listing every identity used in the chapter.
- Solutions PDF separate: The matrices 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 chapter notes 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 Matrices Chapter
The most repeated question patterns in CBSE Class 12 Maths for the Matrices 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 matrices class 12 important questions you will see on board day.
- matrices class 12 previous year questions for 2019-2024 are linked from the PYQ block at the bottom of this page - the exact CBSE phrasings.
- The matrices class 12 important questions with solutions set is reused by toppers in the last fortnight of revision.
- For NCERT Exemplar practice, the matching matrices 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 Matrices 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 matrices class 12 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 Matrices Notes Pair with NCERT Solutions and the Formula Sheet
The the PDF 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 |
|---|---|---|
| Matrices Notes (this page) | Theory, definitions, exam patterns | First pass, before practice |
| the PDF PDF | Step-by-step solved exercises | Second pass, during NCERT practice |
| this chapter 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 this chapter cover every back-of-chapter exercise plus the miscellaneous exercise.
- The matrices 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 these notes formulas reference sheet is the same A4 file students sometimes refer to as this Class 12 page all formulas - it lists every identity used in the chapter.
- State-board references: RD Sharma, ML Aggarwal, Teachoo and the Maharashtra board the resource textbook PDF all share the same core definitions.
- For class-first search phrasings - class 12 matrices solutions, class 12 matrices ncert solutions, ncert class 12 matrices 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 chapter notes |
|---|---|
| RD Sharma Class 12 Matrices | Question patterns overlap with NCERT at ~70%; an advanced supplement. |
| ML Aggarwal Class 12 Matrices | Solutions style is closer to JEE; good for problem-solving practice. |
| Teachoo the PDF | Free online walkthroughs; useful for video-style learning. |
| Shaalaa matrices class 12 solutions | State-board (Maharashtra HSC) phrasings; same core definitions. |
| Maharashtra board this chapter textbook PDF | Same chapter content under the HSC syllabus; exercise numbers differ. |
| NCERT Exemplar Class 12 Matrices | Advanced problems for JEE Main/JEE Advanced preparation. |
How to Use the Matrices Notes Page Most Effectively
The recommended study plan for these notes 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 these notes 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 matrices class 12 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.
Class 12 Mathematics Revision Strategy and Exam Practice Routines
Most CBSE Class 12 students benefit from a three-pass revision rhythm: the first pass is slow and definition-by-definition, the second works through every back-of-chapter problem, and the third uses past board papers at exam pace. JEE and CUET aspirants should add a fourth pass focused on the JEE-specific question bank, because the same chapter content gets tested under different time pressure. Within these passes, a few habits separate students who hit the 85+ band from the rest:
- Read two previous-year marking schemes before the exam — marking-scheme phrasings reward exact wording, which pays off more than another mock paper.
- Write a one-page formula recall sheet per chapter that fits on one side of A4; the night before the exam should be spent only on this sheet and a single full-length mock.
- Solve the CBSE 2026-27 sample paper twice — it is the highest-fidelity guide to question difficulty and lifts mock-paper accuracy by 8 to 12 percent.
- Self-evaluate every two hours by writing the chapter's key results from memory, rather than reading passively.
- Finish back-of-chapter exercises once and revisit the miscellaneous exercise twice — past-board data shows this is worth roughly 2 extra marks.
Common arithmetic slips cost most students at least one mark per paper, and most marks lost in long-answer questions go to incomplete working, not wrong answers. Write every intermediate step in full, even on questions that feel straightforward — method marks are claimed step by step even when the final number is off. The case-study format introduced in recent CBSE boards now appears regularly, framing a real-world scenario that tests definitions plus one-step applications, so practising case studies from the CBSE sample paper translates directly into marks.
Time allocation in the last fortnight matters most. Two thirds of revision time should go to weak chapters, the remaining third to maintaining strong ones; students who revise this chapter twice in the last 10 days score 1.5 to 2 marks higher on past boards. The night before the exam is best spent on:
- The one-page formula recall sheet built earlier in revision.
- A single full-length mock paper at exam timing.
- Avoid learning any new material the night before — sleep matters more.
Mock papers serve two distinct purposes — subject mocks build chapter-level recall while full-paper mocks build time-management discipline. Tracking your own mock-paper scores week by week is the single best predictor of board outcome; a simple spreadsheet with date, paper, score, and one note on a recurring mistake is enough. For students using only one reference, the printed NCERT remains the highest-yield resource — books beyond NCERT add depth but rarely change board outcomes, since the marking scheme rewards NCERT phrasing first. Hindi-medium students can keep the bilingual NCERT edition handy because it follows the same notation, and group study works best when each student picks one sub-topic to explain.
Past CBSE marking schemes from 2020 to 2024 show that average board marks for Class 12 Maths have settled around the 75 to 82 percent band. Students who hit the upper end usually share the same revision rhythm: NCERT first, mock papers second, and previous-year papers third.
Matrices Class 12 NCERT Solutions - Frequently Asked Questions
Ques. Where can I download the chapter notes for free?
Ans. You can download the NCERT Solutions for Class 12 Maths Chapter 3 Matrices Exercise 3.4 PDF directly from the the PDF. Both the Normal and HD versions are available, and both are free.
Ques. How many questions are there in Class 12 Maths Exercise 3.4?
Ans. Exercise 3.4 of Class 12 Maths Chapter 3 Matrices contains 18 questions covering inverse of a matrix by elementary row and column operations. Questions 1 to 6 deal with 2 by 2 matrices, questions 7 to 17 with 3 by 3 matrices, and question 18 is a multiple-choice item on when an inverse exists.
Ques. What is the main concept tested in Class 12 Maths Chapter 3 Exercise 3.4?
Ans. Exercise 3.4 tests the inverse of a square matrix using elementary transformations, that is row operations applied to A = IA or column operations applied to A = AI . The exercise requires students to drive the left side to the identity matrix, after which the right side becomes A-1 .
Ques. Is this this chapter aligned with the 2026-27 NCERT syllabus?
Ans. Yes. This page hosts these notes, and reflects the current 2026-27 syllabus for Class 12 Mathematics. Exercise 3.4 has been kept intact in the new NCERT edition and all 18 questions appear in the current textbook.
Ques. How many pages is the Class 12th Maths Matrices Exercise 3.4 Solutions PDF?
Ans. The NCERT Solutions PDF for Exercise 3.4 runs approximately 28 pages and covers every question with step-by-step working, the elementary operation symbol at each step, and a verification line at the end.
Ques. How do you find the inverse of a 3 by 3 matrix using elementary operations?
Ans. Start by writing A = IA , where A is the given 3 by 3 matrix and I is the 3 by 3 identity. Apply elementary row operations in a fixed order:
first make the (1,1) entry equal to 1, then zero out the rest of column 1, then make (2,2) equal to 1, zero out the rest of column 2, and finally make (3,3) equal to 1 and zero out column 3. When the left side becomes the identity matrix, the right side is A-1 .
Ques. Can I use column operations instead of row operations in Exercise 3.4?
Ans. Yes, but you must start with A = AI (not A = IA ) and apply column operations only. Mixing row and column operations in the same solution gives an incorrect inverse. CBSE marking schemes accept either method, but most students find row operations faster for the 3 by 3 matrices that cover Exercise 3.4.
Ques. What happens if a row of zeros appears while solving Exercise 3.4 questions?
Ans. A full row of zeros on the left side during the working means the matrix is singular, that is its determinant is zero, and the inverse does not exist. Questions 15 and 16 of Exercise 3.4 are designed to test this case, and the correct answer is to state that the inverse does not exist (do not continue trying to invert).
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