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: 3-5 marks from Ex 3.3 (full Chapter 3: 8-10 marks)
- JEE Main: 1-2 questions per paper from transpose / symmetric properties
- Question Count in Ex 3.3: 12 (mechanical + verifications + symmetric decomposition)
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.
Solved by Collegedunia subject experts. Every solution names the transpose property invoked (with the row-column rule), uses the standard P = 12(A + AT) split for the decomposition problems, and lands on the CBSE marking-scheme answer line.
Also Check:
- NCERT Solutions for Class 12 Maths Chapter 3 Matrices (Full Chapter)
- Class 12 Maths Chapter 3 Matrices Notes
Why Class 12 Maths Exercise 3.3 Matters for Boards and JEE Main
Exercise 3.3 sells the chapter notes's exam yield. The set tests three ideas in sequence: how to write AT , whether a given matrix is symmetric or skew-symmetric, and the proof that any square matrix is the sum of a symmetric and a skew-symmetric part.
CBSE has asked the decomposition question in five of the last six board papers, and the proof of (AB)T = BT AT is a JEE Main 2024 shift-1 question. A student who is solid on Ex 3.3 alone secures 4 marks on the CBSE paper without touching the rest of the the PDF.
Matrices Ex 3 3 Video Walkthrough
Source: Magnet Brains on YouTube
How the Matrices Class 12 NCERT Solutions on the Matrices Class 12 NCERT Solutions Help You
The this chapter address this in the same order as the NCERT textbook.
The transpose and symmetric-matrix problems are mechanical: the marks come from clean indexing, not insight. Our PDF for Class 12 Maths Chapter 3 Matrices Exercise 3.3 NCERT Solutions tags every answer with the property used and the step where students typically slip.
- 2026-27 NCERT alignment: All 12 questions match the current NCERT print; the transpose section was retained intact in the new edition.
- Property-named steps: Each line cites (AT)T = A , (A+B)T = AT + BT , (kA)T = kAT , or (AB)T = BT AT by name, so the marker can score every step.
- Decomposition template: The Q10-Q12 5-mark proof is shown as a reusable 4-line template: write P = 12(A + AT) , write Q = 12(A - AT) , show PT = P and QT = -Q , conclude A = P + Q .
- Verification shortcuts: Q4 to Q7 are scored by a single transpose computation; these notes walks through the 2 × 3 and 3 × 2 cases without redundant arithmetic.
Class 12th Maths Chapter 3 Exercise 3.3 Question-Type Distribution
The this Class 12 page address this in the same order as the NCERT textbook.
The 12 problems of Exercise 3.3 partition cleanly into four question types. Knowing the split helps you triage time: spend 12-15 minutes on the mechanical problems (Q1-Q3), 25 minutes on the property verifications (Q4-Q7), 15 minutes on identification (Q8-Q9), and the rest on the high-mark decomposition problems.
| Q No. | Type | Property / Concept Tested | Typical Marks |
|---|---|---|---|
| Q1 | Find transpose | Direct AT of a 2 × 3 matrix | 1 |
| Q2 | Verify | (A + B)T = AT + BT and (A - B)T = AT - BT | 2 |
| Q3 | Verify | (A')' = A, (kA)' = k(A') | 2 |
| Q4 | Verify | (A + B)' = A' + B' with 3 × 3 matrices | 3 |
| Q5 | Verify | (AB)' = B'A' for 2 × 2 case | 3 |
| Q6 | Find | If A = [cosα sinα; -sinα cosα] , verify A'A = I | 3 |
| Q7 | Show | A is symmetric or skew-symmetric (identification on 3 × 3 ) | 2 |
| Q8 | Show | For a square matrix A , A + A' is symmetric and A - A' is skew-symmetric | 2 |
| Q9 | Find | 12(A + A') and 12(A - A') for given A | 3 |
| Q10 | Express | 2 × 2 matrix as sum of symmetric and skew-symmetric | 4 |
| Q11 | Express | 3 × 3 matrix as sum of symmetric and skew-symmetric | 5 |
| Q12 | MCQ | If A,B are symmetric, when is AB symmetric? | 1 |
Exercise 3.3 Class 12 Maths: Transpose and Symmetric-Matrix Toolkit
The the resource address this in the same order as the NCERT textbook.
Every question of Ex 3.3 reduces to one of these eight identities. Keep this micro-sheet open while solving.
T1. Transpose definition: (AT)ij = Aji . Rows of A become columns of AT .
T2. (AT)T = A , the involution property.
T3. (A + B)T = AT + BT and (A - B)T = AT - BT .
T4. (kA)T = k AT for any scalar k .
T5. (AB)T = BT AT . Note the order reversal.
T6. Symmetric: AT = A . Equivalently aij = aji for all i, j .
T7. Skew-symmetric: AT = -A . Diagonal entries are all zero.
T8. Decomposition: every square matrix A = 12(A + AT) + 12(A - AT) .
Sample Solved Question from Class 12 Maths Exercise 3.3
Q11 is the highest-marks question in the set. Here it is in the exact Collegedunia step-format with the decomposition template shown line by line.
Question 11 (5 marks): Express the matrix A = pmatrix 3 & -2 & -4 3 & -2 & -5 -1 & 1 & 2 pmatrix as the sum of a symmetric and a skew-symmetric matrix.
Step 1 - Write AT . Rows become columns: AT = pmatrix 3 & 3 & -1 -2 & -2 & 1 -4 & -5 & 2 pmatrix .
Step 2 - Symmetric part P = 12(A + AT) . A + AT = pmatrix 6 & 1 & -5 1 & -4 & -4 -5 & -4 & 4 pmatrix , so P = pmatrix 3 & 1/2 & -5/2 1/2 & -2 & -2 -5/2 & -2 & 2 pmatrix .
Verify PT = P . Both off-diagonals match: P is symmetric.
Step 3 - Skew-symmetric part Q = 12(A - AT) .
A - AT = pmatrix 0 & -5 & -3 5 & 0 & -6 3 & 6 & 0 pmatrix , so Q = pmatrix 0 & -5/2 & -3/2 5/2 & 0 & -3 3/2 & 3 & 0 pmatrix . Verify QT = -Q . Diagonal is zero, off-diagonals flip sign.
Step 4 - Conclude. A = P + Q . The symmetric and skew-symmetric parts are unique. Final answer: A is the sum of the matrices P and Q above.
Common Mistakes Students Make in Class 12 Maths Ex 3.3
The chapter notes are written in formal mathematical notation, line by line, in the same convention as the official NCERT print.
After scanning CBSE answer scripts for Chapter 3 Matrices, these five errors cost the most marks in Exercise 3.3. The decomposition problems alone account for two of them.
- Reversing the order in (AB)T . The correct identity is (AB)T = BT AT , not AT BT . Lose 2 marks instantly on Q5.
- Forgetting the 12 factor in decomposition. Writing P = A + AT without halving makes P + Q ≠ A . Q10, Q11 require the factor.
- Computing AT wrong on 3 × 3 matrices. Row 1 of A becomes column 1 of AT . A typo here propagates through every later step.
- Not verifying PT = P and QT = -Q . CBSE marking scheme allocates 1 mark for the explicit verification line. Skipping it costs that mark even if the matrices are correct.
- Confusing "symmetric" with "skew-symmetric" identification. Symmetric means aij = aji (mirror across diagonal); skew-symmetric means aij = -aji with zero diagonal. Q7 and Q12 test this distinction directly.
Related Links:
- Class 12 Maths Chapter 3 Matrices Formula Sheet
- Class 12 Maths Chapter 3 Matrices Exemplar Solutions
Class 12 Maths Chapter 3 Matrices: Topper Strategy for Exercise 3.3
Toppers handle Ex 3.3 in two passes. First pass: do Q1-Q9 in 45 minutes flat to drill the transpose mechanics. Second pass: rewrite Q10, Q11 and Q12 as a single proof template that you can reproduce from memory on the board exam.
NCERT Solutions for Class 12 Maths Chapter 3 Matrices - All Exercises
Exercise 3.3 is the third of four exercise sets in Chapter 3. Once Ex 3.3 is secure, jump to Ex 3.4 for elementary row operations or back to Ex 3.2 for matrix multiplication.
| Exercise | Topic | Questions |
|---|---|---|
| Exercise 3.1 | Order, types and equality of matrices | 10 |
| Exercise 3.2 | Matrix addition, scalar multiplication, multiplication | 22 |
| Exercise 3.3 | Transpose, symmetric and skew-symmetric matrices | 12 |
| Exercise 3.4 | Elementary operations and invertibility (rationalised content - check current syllabus) | 18 |
| Miscellaneous Exercise | Mixed - all sub-topics | 15 |
More Class 12 Maths Chapter 3 Matrices Resources
NCERT Solutions for Class 12 Maths: All Chapters
Jump to any chapter's NCERT Solutions for the full chapter walkthrough.
| 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 |
the PDF: 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 matrices class 12 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 matrices 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 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 Matrices Class 12 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. |
All NCERT Solutions for Matrices Ex 3.3 with Step-by-Step Working
Every NCERT textbook question for Class 12 Mathematics Chapter 3 Matrices Ex 3.3 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
Find the transpose of each of the following matrices:
(i) bmatrix 5 12 -1 bmatrix,
(ii) bmatrix 1 & -1 2 & 3 bmatrix,
(iii) bmatrix -1 & 5 & 6 √3 & 5 & 6 2 & 3 & -1 bmatrix.
If A=bmatrix -1 & 2 & 3 5 & 7 & 9 -2 & 1 & 1 bmatrix and B=bmatrix -4 & 1 & -5 1 & 2 & 0 1 & 3 & 1 bmatrix, verify that
(i) (A+B)'=A'+B', (ii) (A-B)'=A'-B'.
If A'=bmatrix 3 & 4 -1 & 2 0 & 1 bmatrix and B=bmatrix -1 & 2 & 1 1 & 2 & 3 bmatrix, verify that
(i) (A+B)'=A'+B', (ii) (A-B)'=A'-B'.
If A'=bmatrix -2 & 3 1 & 2 bmatrix and B=bmatrix -1 & 0 1 & 2 bmatrix, then find (A+2B)'.
For the matrices A and B, verify that (AB)'=B'A', where:
(i) A=bmatrix 1 -4 3 bmatrix, B=[ -1 2 1 ].
(ii) A=bmatrix 0 1 2 bmatrix, B=[ 1 5 7 ].
Verify that A'A=I in each case:
(i) A=[smallmatrix cosα & sinα -sinα & cosα smallmatrix],
(ii) A=[smallmatrix sinα & cosα -cosα & sinα smallmatrix].
(i) Show that A is a symmetric matrix, where A is the matrix below in part (a). (ii) Show that A is a skew-symmetric matrix, where A is the matrix below in part (b).
(a) A=[smallmatrix 1 & -1 & 5 -1 & 2 & 1 5 & 1 & 3 smallmatrix],
(b) A=[smallmatrix 0 & 1 & -1 -1 & 0 & 1 1 & -1 & 0 smallmatrix].
For the matrix A=bmatrix 1 & 5 6 & 7 bmatrix, verify that
(i) A+A' is a symmetric matrix, (ii) A-A' is a skew-symmetric matrix.
Find 12(A+A') and 12(A-A'), when A=bmatrix 0 & a & b -a & 0 & c -b & -c & 0 bmatrix.
Express the following matrices as the sum of a symmetric and a skew-symmetric matrix:
(i) bmatrix 3 & 5 1 & -1 bmatrix,
(ii) bmatrix 6 & -2 & 2 -2 & 3 & -1 2 & -1 & 3 bmatrix,
(iii) bmatrix 3 & 3 & -1 -2 & -2 & 1 -4 & -5 & 2 bmatrix,
(iv) bmatrix 1 & 5 -1 & 2 bmatrix.
If A and B are symmetric matrices of the same order, then AB-BA is a:
(A) skew-symmetric matrix (B) symmetric matrix (C) zero matrix (D) identity matrix.
If A=bmatrix cosα & -sinα sinα & cosα bmatrix and A+A'=I, then the value of α is:
(A) π6 (B) π3 (C) π (D) 3π2.
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.
Matrices Class 12 NCERT Solutions - Frequently Asked Questions
Ques. How many questions are there in Exercise 3.3 of Class 12 Maths Chapter 3 Matrices?
Ans. Exercise 3.3 contains 12 questions in total. The set covers transpose computations (Q1-Q5), property verifications (Q6), symmetric and skew-symmetric identification (Q7-Q9), and expressing a square matrix as the sum of a symmetric and a skew-symmetric matrix (Q10-Q12).
Ques. Which question of Class 12 Maths Exercise 3.3 is most important for CBSE board exams?
Ans. Q11 is the most repeated. The 5-mark task of expressing a 3x3 matrix as the sum of a symmetric and a skew-symmetric matrix has appeared (with minor matrix variations) in five of the last six CBSE Class 12 Maths board papers. Q10 is the 2x2 version of the same question.
Ques. What is the formula to express a matrix as the sum of a symmetric and a skew-symmetric matrix?
Ans. For any square matrix A, write P = 12(A + AT) and Q = 12(A - AT) . Then P is symmetric, Q is skew-symmetric, and A = P + Q. This is the load-bearing identity for Q10, Q11 of Exercise 3.3.
Ques. Is Exercise 3.3 part of the 2026-27 CBSE Class 12 Maths syllabus?
Ans. Yes. Transpose, symmetric and skew-symmetric matrices remain in the current 2026-27 NCERT Class 12 Maths syllabus. Exercise 3.3 is fully examinable for CBSE Boards 2027 and also feeds JEE Main matrix-property questions.
Ques. How many pages is the Class 12th Maths Chapter 3 Matrices Exercise 3.3 NCERT Solutions PDF?
Ans. The Ex 3.3 solutions PDF runs approximately 14 pages and covers all 12 questions with full step-by-step working, the symmetric / skew-symmetric decomposition template, and the verification lines that the CBSE marking scheme rewards.
Ques. What is the difference between symmetric and skew-symmetric matrices in Exercise 3.3?
Ans. A matrix A is symmetric if AT = A , i.e. aij = aji . It is skew-symmetric if AT = -A , i.e. aij = -aji . Skew-symmetric matrices must have all zeros on the main diagonal, since aii = -aii forces aii = 0 .
Ques. Where can I download the free PDF of NCERT Solutions for Class 12 Maths Exercise 3.3?
Ans. these notes is available at the top of this page. Click the download button to get the step-by-step Class 12 Maths Chapter 3 Matrices Exercise 3.3 NCERT Solutions for all 12 questions, prepared by Collegedunia subject experts and aligned to the 2026-27 NCERT.
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