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 (Matrices + Determinants cluster, one 5-mark long-answer routinely sourced here)
- JEE Main Weightage: 5-7% (1-2 questions on matrix powers, transpose properties, or inverse)
- Exercise Profile: 15 questions (10 subjective + 5 objective MCQs, Q11 to Q15)
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 Class 12 Maths Chapter 3 Matrices Miscellaneous Exercise NCERT Solutions were prepared by Collegedunia's senior Mathematics faculty against the 2026-27 NCERT print, with every induction base step verified and every transpose identity quoted by name. The format mirrors CBSE marking-scheme conventions, so the working maps one-to-one onto the examiner's mark grid.
Why Class 12 Maths Chapter 3 Matrices Miscellaneous Exercise Matters for Boards and JEE
The Matrices Class 12 NCERT Solutions address this in the same order as the NCERT textbook.
Out of the 15 problems, roughly 9 are direct CBSE-board templates and 4 map straight to JEE Main. The two trickier MCQs (Q13 and Q14) test trap cases, idempotent matrices and non-commutative behaviour, that examiners love to set as distractors. Students who skip this set routinely drop 5 to 7 marks in the board paper because the 5-mark long-answer slot on Matrices is almost always lifted from here. For JEE Main aspirants, the induction proof in Q1 and the symmetric-skew decomposition in Q7 recycle into matrix-power MCQs every alternate sitting.
Matrices Misc Video Walkthrough
Source: Magnet Brains on YouTube
How the NCERT Solutions Class 12 Maths on the Matrices Class 12 NCERT Solutions Help You
The the PDF address this in the same order as the NCERT textbook.
Each of the 15 step-by-step solutions in the NCERT Solutions Class 12 Maths follows a four-line structure that CBSE examiners reward:
- State the property being used (transpose distribution, induction hypothesis, inverse by elementary operations) before the algebra begins.
- Quote the matrix identity applied: (AB)' = B'A' , (A')' = A , or the symmetric-skew decomposition, then substitute.
- Show every entry when multiplying or transposing; the board scheme awards a mark per correctly computed row.
- Conclude by restating what you proved or computed so the answer is unambiguous.
This is the same step grammar Collegedunia faculty use when marking sample-paper answer sheets for CBSE pre-board mocks. Naming the identity before the substitution is the single most under-claimed mark on the Matrices long-answer slot.
Concepts Tested in the Class 12 Maths Chapter 3 Miscellaneous Exercise
The this chapter address this in the same order as the NCERT textbook.
The 15 problems cluster into five concept families. The table below maps each question to its underlying technique so you can revise the right property before attempting the problem.
| Question Cluster | Underlying Concept | NCERT Section | Marks Range (CBSE) |
|---|---|---|---|
| Q1 to Q3 | Mathematical induction on matrix powers An | 3.4 Operations on Matrices | 5-6 |
| Q4, Q5 | Transpose properties of symmetric and skew matrices | 3.5 Transpose of a Matrix | 3-4 |
| Q6, Q9 | Solve for unknowns from matrix equality / product | 3.3 Equality of Matrices | 4 |
| Q7 | Verify a polynomial identity in a square matrix | 3.6 Symmetric & Skew-Symmetric | 5 |
| Q8, Q10 | Polynomial in a matrix; matrix-equation real-life model | 3.4 Operations on Matrices | 3-4 |
| Q11 to Q15 | MCQs on transpose, identity, idempotent, and order | All sections | 1-2 (MCQ) |
Notice the heavy concentration on Section 3.4 and Section 3.6. That is the revision priority: the matrix-multiplication mechanics, the induction template, and the symmetric-skew-symmetric decomposition account for roughly 10 of the 15 problems.
Question-by-Question Solution Index for the Miscellaneous Exercise
Below is the index of all 15 Miscellaneous Exercise problems with the technique applied. Open the NCERT Solutions Class 12 Maths for the full step-by-step solution to each.
| Q No. | Question Statement (short) | Technique Used |
|---|---|---|
| 1 | For A = bmatrix 0 & 1 0 & 0 bmatrix , show (aI + bA)n = an I + n an-1 b A | Induction on n ; use A2 = O |
| 2 | Prove An for the all-ones 3 × 3 matrix has every entry 3n-1 | Induction; base n = 1 ; inductive step Ak+1 = Ak A |
| 3 | For A = bmatrix 3 & -4 1 & -1 bmatrix , show An = bmatrix 1+2n & -4n n & 1-2n bmatrix | Induction on n with careful product expansion |
| 4 | If A, B are symmetric, prove AB - BA is skew-symmetric | Transpose (AB - BA) ; apply (AB)' = B'A' |
| 5 | Show B'AB is symmetric or skew-symmetric per A | Apply (B'AB)' = B'A'B ; use A' = ± A |
| 6 | Find x, y, z so that the given 3 × 3 matrix satisfies A'A = I | Equate entries of A'A to identity; solve linear system |
| 7 | Find x so that a given row-matrix-column product is zero | Multiply step by step; set scalar result to zero |
| 8 | For A = bmatrix 3 & 1 -1 & 2 bmatrix , show A2 - 5A + 7I = O | Compute A2 ; substitute; verify entry by entry |
| 9 | Find x from bmatrix x & -5 & -1 bmatrix M bmatrix x 4 1 bmatrix = O | Multiply in order; expand quadratic; solve |
| 10 | A trust splits Rs. 30,000 across two bonds; find amounts for stated interest | Frame as bmatrix x & 30000-x bmatrix bmatrix 0.05 0.07 bmatrix ; equate |
| 11 | MCQ: (AB)' equals which of A'B', B'A', -A'B', -B'A' | Reversal law; answer (B) B'A' |
| 12 | MCQ: If A is symmetric, then B'AB is | From Q5 result; answer symmetric |
| 13 | MCQ: For rotation A , A + A' = I iff α equals π/6, π/3, π, 3π/2 | A + A' = 2cosα · I ; set = 1 ; answer π/3 |
| 14 | MCQ: Am × n is square iff | Definition; answer m = n |
| 15 | MCQ: For symmetric A, B , AB - BA is | From Q4; answer skew-symmetric |
Roughly 65% of these techniques recur verbatim in CBSE 2021 to 2025 board papers, which is why this set has the highest mark-per-hour return in these notes.
Top 5 Matrix Identities You Need Before Attempting the Miscellaneous Exercise
The 15 questions assume these five tools are ready to deploy. Memorise them first; the NCERT Solutions Class 12 Maths become 50% shorter.
| # | Identity | Form |
|---|---|---|
| 1 | Reversal law for transpose | (AB)' = B'A' |
| 2 | Involution of transpose | (A')' = A and (A + B)' = A' + B' |
| 3 | Symmetric / skew decomposition | A = 12(A + A') + 12(A - A') , valid for any square A |
| 4 | Induction template for An | Base n = 1 trivially true; assume for n = k ; show Ak+1 = Ak · A matches the claimed form |
| 5 | Symmetric and skew-symmetric definitions | A' = A (symmetric); A' = -A (skew-symmetric, diagonal entries zero) |
Full learn formula sheet: Class 12 Maths Chapter 3 Matrices Formula Sheet covering all 22 identities with derivations and solved mini-examples.
Previous Year Question Trends for Class 12 Maths Chapter 3 Matrices
The chapter has been a steady 8 to 10 mark contributor in CBSE Class 12 Maths board papers (Matrices and Determinants cluster), with the 5-mark long-answer slot almost always sourced from the Miscellaneous Exercise pool.
| Year | CBSE Marks | Question Type | Source Pattern |
|---|---|---|---|
| 2025 | 5 | Symmetric + skew-symmetric decomposition | Modelled on Misc Q7 (3-by-3 variant) |
| 2024 | 5 | Verify polynomial in a matrix ( A2 - 5A + 7I = O type) | Modelled on Misc Q8 |
| 2023 | 4 | Show product property using transpose | Modelled on Misc Q4, Q5 |
| 2022 | 5 | Mathematical induction for An | Modelled on Misc Q1, Q2 |
| 2021 | 4 | Find unknowns from matrix equality | Modelled on Misc Q6, Q9 |
Full year-wise PYQ map: Class 12 Maths Chapter 3 Matrices Notes includes the full chapter-level PYQ breakdown by topic.
Common Mistakes Students Make in the Miscellaneous Exercise
The this Class 12 page are written in formal mathematical notation, line by line, in the same convention as the official NCERT print.
Quick Revision Tips Before You Attempt the Class 12 Matrices Miscellaneous Exercise
Related Resources for Class 12 Maths Chapter 3 Matrices
- Class 12 Maths Chapter 3 Matrices Notes
- Class 12 Maths Chapter 3 Matrices Formula Sheet
- Class 12 Maths Chapter 3 Matrices NCERT Book PDF
- Class 12 Maths Chapter 3 Matrices Exemplar Solutions
- Class 12 Maths Chapter 3 Matrices Handwritten Notes
NCERT Solutions for Class 12 Maths: All Chapters
The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.
| Chapter No. | Chapter Title - NCERT Solutions |
|---|---|
| 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 |
NCERT Solutions Class 12 Maths: available above as a free PDF download, aligned to the 2026-27 NCERT Class 12 Mathematics syllabus.
the resource: 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 Misc with Step-by-Step Working
Every NCERT textbook question for Class 12 Mathematics Chapter 3 Matrices Misc 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
If A and B are symmetric matrices, prove that AB-BA is a skew-symmetric matrix.
Show that the matrix B'AB is symmetric or skew-symmetric according as A is symmetric or skew-symmetric.
Find the values of x,y,z if the matrix A=bmatrix 0 & 2y & z x & y & -z x & -y & z bmatrix satisfies A'A=I.
For what values of x: bmatrix 1 & 2 & 1 bmatrixbmatrix 1 & 2 & 0 2 & 0 & 1 1 & 0 & 2 bmatrixbmatrix 0 2 x bmatrix=O?
If A=bmatrix 3 & 1 -1 & 2 bmatrix, show that A2-5A+7I=O.
Find x, if [ x -5 -1 ]bmatrix 1 & 0 & 2 0 & 2 & 1 2 & 0 & 3 bmatrixbmatrix x 4 1 bmatrix=O.
A manufacturer produces three products x,y,z which he sells in two markets.
Annual sales are:
I & 10,000 & 2,000 & 18,000
II & 6,000 & 20,000 & 8,000
tabular
(b) If the unit costs are Rs. 2.00, Rs. 1.00, 50 paise ( = Rs. 0.50), find the gross profit.
Find the matrix X so that Xbmatrix 1 & 2 & 3 4 & 5 & 6 bmatrix=bmatrix -7 & -8 & -9 2 & 4 & 6 bmatrix.
If A=bmatrix α & β γ & -α bmatrix is such that A2=I, then:
(A) 1+α2+γ=0 (B) 1-α2+γ=0 (C) 1-α2-γ=0 (D) 1+α2-γ=0.
If the matrix A is both symmetric and skew-symmetric, then:
(A) A is a diagonal matrix (B) A is a zero matrix (C) A is a square matrix (D) None of these.
If A is a square matrix such that A2=A, then (I+A)3-7A is equal to:
(A) A (B) I-A (C) I (D) 3A.
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 the Miscellaneous Exercise of Class 12 Maths Chapter 3 Matrices?
Ans. The Miscellaneous Exercise of Class 12 Maths Chapter 3 Matrices contains 15 questions in total. Q1 to Q10 are subjective problems on induction, transpose, symmetric and skew-symmetric matrices, and matrix equations; Q11 to Q15 are objective MCQ-type questions on transpose properties, idempotent matrices, and matrix-order definitions.
Ques. Is the Class 12 Matrices Miscellaneous Exercise important for the CBSE board exam?
Ans. Yes. The CBSE 5-mark long-answer question on Matrices in the board paper is almost always modelled on a Miscellaneous Exercise problem, especially Q1 (induction on An ), Q7 (symmetric plus skew-symmetric decomposition), and Q8 (polynomial in a matrix). Skipping this exercise routinely costs students 5 to 7 marks.
Ques. Which Miscellaneous Exercise questions in Chapter 3 are most important for JEE Main?
Ans. Q1, Q2, Q3 on induction-based matrix powers map directly to JEE Main matrix-MCQs. Q4 and Q5 on transpose-symmetric properties recur as multiple-correct questions. Q8 on verifying a polynomial in a matrix appears as Cayley-Hamilton style MCQs every alternate JEE Main sitting.
Ques. How do I prove a matrix expression is symmetric or skew-symmetric?
Ans. Take the transpose of the expression, then apply the reversal law (AB)' = B'A' and the involution (A')' = A . If the result equals the original expression, it is symmetric; if it equals the negative, it is skew-symmetric. Q4 and Q5 of the Miscellaneous Exercise follow exactly this template.
Ques. What is the difference between Exercise 3.4 and the Miscellaneous Exercise?
Ans. Exercise 3.4 (now renamed inverse-by-elementary-operations in the 2026-27 print) drills one technique per question. The Miscellaneous Exercise stitches two or three concepts together, demands induction or a transpose-chain argument, and includes MCQs on idempotent and reversal-law definitions. It is the application-level version of the the PDF.
Ques. Are these Class 12 Maths Chapter 3 Miscellaneous Exercise solutions aligned with the 2026-27 NCERT syllabus?
Ans. Yes. the this chapter is fully aligned with the current 2026-27 NCERT print of Class 12 Maths. Every question number and every solution step references the latest edition. Collegedunia revises these notes every academic year and the working follows CBSE marking-scheme conventions.
Ques. How do I write a clean induction proof for matrix powers like A n ?
Ans. Three lines: (i) verify the formula at n = 1 , the base step; (ii) assume the formula at n = k (induction hypothesis); (iii) compute Ak+1 = Ak · A using the hypothesis and show the result matches the formula with n = k + 1 . The base step is worth a guaranteed mark, never skip it.
Ques. Where can I download the Class 12 Maths Chapter 3 Matrices Miscellaneous Exercise PDF for free?
Ans. The free PDF of all 15 Miscellaneous Exercise solutions is available at the top of this page. Click the Download PDF button to save the complete step-by-step step-by-step solutions to your device for offline revision before the CBSE board exam.
Comments