Matrices Class 12 Notes - Class 12 Maths Ch 3

The Matrices Class 12 Notes hosted here summarise Class 12 Mathematics Chapter 3 Matrices in the order taught by the NCERT. Each result in the the resource is stated formally, then explained in a single short paragraph, then illustrated with one solved examples. The notes keep the notation of the PDF so cross-referencing remains direct.

• CBSE Boards: 6 to 8 marks (Algebra unit with Determinants gives 10 to 13 marks)
• JEE Main: Matrices and Determinants together carry 2 to 3 questions, about 6 to 8% of the Maths section
• CUET (UG) Maths: 2 to 3 direct MCQs on multiplication, transpose, and inverse via EROs every year

If you want to lock in every rule of matrix algebra before walking into the exam hall, the chapter notes below give you precise definitions, solved formats, and the elementary-row-operation drill for finding an inverse, exactly the format CBSE and JEE Main test in the Algebra unit.

These Collegedunia the the PDF are compiled by Class 12 Maths specialists and cross-checked against the 2026-27 NCERT textbook and the past five years of CBSE marking schemes, so what you revise lines up with what examiners actually award marks for.

Class 12 Maths Chapter 3 Matrices: Topic-by-Topic Breakdown

Matrices is genuinely an operations-learn chapter. Once the rules of addition, multiplication, transpose, and inverse click, the questions become procedural. Here is the walkthrough, in the order the NCERT chapter introduces each concept, with the formulas you must memorise.

1. What is a matrix? Order, elements, and equality

A matrix is an ordered rectangular array of numbers (or functions) arranged in rows and columns. A matrix with m rows and n columns is said to be of order m × n and contains mn elements in total. The general element in the ith row and jth column is written a_ij, and the matrix as A = [a_ij]_{m × n}.

Two matrices are equal only when they have the same order and every corresponding entry matches, i.e. a_ij = b_ij for all i, j. This equality definition is the basis of "find x and y" problems where you equate corresponding entries.

Quick Note: If a matrix has order m × n and the product mn = 24, then the possible orders are 1 × 24, 2 × 12, 3 × 8, 4 × 6 and their transposes, a classic 1-mark MCQ trap.

2. Types of matrices you must recognise on sight

NCERT lists the types you should identify instantly: row matrix (1 × n), column matrix (m × 1), square matrix (n × n), diagonal matrix (off-diagonal entries are zero), scalar matrix (diagonal entries all equal), identity matrix I_n (scalar with all diagonal entries 1), zero or null matrix O, and upper/lower triangular matrices.

For a square matrix of order n, the diagonal entries are a₁₁, a₂₂, , a_nn. Knowing these classifications is what lets you spot, for instance, that the only matrix which is both symmetric and skew-symmetric is the zero matrix, a recurring 1-mark question.

3. Matrix addition, subtraction, and scalar multiplication

4. Matrix multiplication: compatibility and non-commutativity

5. Transpose of a matrix and its properties

6. Symmetric and skew-symmetric matrices

7. Elementary row and column operations

8. Invertible matrices and inverse by EROs: the JEE extension

Matrices Video Walkthrough

Source: Magnet Brains on YouTube

How will Collegedunia's NCERT Notes for Class 12 Maths Chapter 3 help you?

• Every definition (row, column, square, diagonal, scalar, identity, zero, symmetric, skew-symmetric matrices) is given with an example in the exact NCERT order, so cross-referencing is painless.
• Matrix multiplication is broken down with a compatibility check (m × n by n × p) and a solved dot-product layout, the way CBSE expects the steps in a 3-mark question.
• The symmetric-plus-skew-symmetric decomposition is shown as a templated 4-step process, since this is a recurring 4-mark question.
• Inverse using elementary row operations is treated as the JEE-extension topic, with annotated row-reductions and the singular-matrix flag that examiners love to plant.

Class 12 Maths Chapter 3 Matrices: Most Important Topics

The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.

Matrices Class 12 Notes: Most Repeated Questions in Board Exams

Ques. If A is a square matrix such that A² = A, find (I + A)³ - 7A. (2024, 2023, 2020)

Ques. Express A = bmatrix3 & -2 & -4 3 & -2 & -5 -1 & 1 & 2bmatrix as the sum of a symmetric and a skew-symmetric matrix. (2024, 2022, 2019)

Ques. Find the inverse of A = bmatrix2 & 1 1 & 1bmatrix using elementary row operations. (2024, 2021, 2018)

Ques. Show that B^T A B is symmetric or skew-symmetric depending on whether A is symmetric or skew-symmetric. (2023, 2020)

Matrices Notes: Previous Year Paper Analysis

• Every paper carries at least one MCQ on order of a matrix, equality, or counting possible orders given a fixed number of elements.
• The symmetric-plus-skew-symmetric decomposition has appeared as a 4-mark question in 4 of the last 6 papers, treat it as a guaranteed question.
• Inverse using elementary row operations is the most predictable 3-mark question, with 2 × 2 and occasional 3 × 3 matrices.
• Short-answer questions on transpose properties, especially (AB)^T = B^T A^T, appear roughly every other year.

Weightage of Matrices in JEE Main, CUET, and CBSE 2026

• In JEE Main 2025, expect 1 to 2 MCQs that often combine matrices with determinants or systems of linear equations.
• CBSE 2025 consistently allotted 6 to 8 marks to Matrices alone, and 10 to 13 marks when combined with Determinants.
• CUET 2025 Maths asked 2 to 3 direct MCQs on matrix multiplication, transpose, and inverse via EROs.

• NCERT Solutions for Class 12 Maths Chapter 3 Matrices
• Class 12 Maths Chapter 3 Matrices Formula Sheet
• Class 12 Maths Chapter 3 Matrices Handwritten Notes

Common Mistakes Students Make in Class 12 Matrices

• Skipping the compatibility check. Writing AB when the column count of A does not match the row count of B. Always state the orders before computing the product.
• Assuming AB = BA. Matrix multiplication is non-commutative. Treating (A + B)² as A² + 2AB + B² is wrong, the correct expansion is A² + AB + BA + B².
• Wrong order in transpose of a product. Writing (AB)^T = A^T B^T instead of (AB)^T = B^T A^T. The reversal is mandatory.
• Computing 12(A + A^T) wrong. Adding instead of averaging, or forgetting the 12 factor entirely, which makes both P and Q double what they should be.
• Mixing row and column operations while finding an inverse. Pick row operations on [A I] OR column operations on bmatrixA Ibmatrix, never both in the same problem.
• Forgetting that AB = O does not mean A = O or B = O. Examples like bmatrix1 & 0 0 & 0bmatrixbmatrix0 & 0 0 & 1bmatrix = O show the zero-product property fails.

NCERT Notes 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.

Matrices Class 12 Notes - Quick Summary

Exercise-wise Breakdown of the Matrices Chapter

The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.

PDF Download Formats and Languages for the Matrices Chapter

The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.

• 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 matrices class 12 pdf - only the exercise numbers differ.

Important Questions and Previous Year Trends for the Matrices Chapter

The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.

• 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

How the Matrices Notes Pair with NCERT Solutions and the Formula Sheet

The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.

• The matrices class 12 ncert solutions 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 matrices class 12 formulas reference sheet is the same A4 file students sometimes refer to as matrices class 12 all formulas - it lists every identity used in the chapter.
• State-board references: RD Sharma, ML Aggarwal, Teachoo and the Maharashtra board matrices class 12 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

How to Use the Matrices Notes Page Most Effectively

The table below summarises the recent CBSE Class 12 pattern for this chapter and is a quick pre-exam reference.

• 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.