📊 CBSE · Class 12 · Mathematics · Chapter 3

Chapter 3
Matrices

Complete chapter resources for CBSE Class 12 Maths — topic breakdown, key formulas, sample questions, previous year board questions, and instant AI question paper generation.

4Topics
8–10Board marks
8Sample questions
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Key Formulas — Chapter 3
  • Matrix addition: (A + B)ᵢⱼ = aᵢⱼ + bᵢⱼ (same order)
  • Transpose: (Aᵀ)ᵢⱼ = aⱼᵢ ; (AB)ᵀ = BᵀAᵀ
  • Symmetric: Aᵀ = A ; Skew-sym: Aᵀ = −A
  • Sym + Skew decomp: A = ½(A+Aᵀ) + ½(A−Aᵀ)
  • Inverse (2×2): A⁻¹ = (1/|A|) · adj A
  • Identity: AI = IA = A ; AA⁻¹ = I
Chapter Overview

What this chapter covers

A matrix is a rectangular array of numbers (or functions) arranged in rows and columns, enclosed in brackets. In CBSE Class 12 Mathematics, Chapter 3 introduces the notation, types, and fundamental operations — addition, subtraction, scalar multiplication, and matrix multiplication — along with the conditions under which each operation is defined (matching orders for addition; compatible orders m×n and n×p for multiplication).

A central concept in this chapter is the transpose of a matrix and the classification of square matrices as symmetric (Aᵀ = A) or skew-symmetric (Aᵀ = −A). Every square matrix can be expressed uniquely as the sum of a symmetric and a skew-symmetric matrix — a result that generates high-value board questions consistently. The chapter also covers elementary row and column operations, which are the foundation for computing the inverse of a matrix.

The inverse of a matrix (A⁻¹) exists only when |A| ≠ 0 (a non-singular matrix) and is computed via the adjoint method: A⁻¹ = (adj A) / |A|. Board questions on this chapter span the full marks range — from 2-mark type-identification and property-verification questions to 5-mark problems involving elementary transformations or solving systems of equations using the matrix method.

Topics

What's inside Chapter 3

As per NCERT Class 12 Mathematics (CBSE syllabus)

Topic 1
Types of Matrices & Basic Operations
Row, column, square, zero, identity, diagonal, scalar, triangular matrices. Addition, subtraction, scalar multiplication, and their properties (commutativity, associativity, distributivity).
Topic 2
Matrix Multiplication
Condition: number of columns of A = number of rows of B. Product order (m×n)(n×p) = m×p. Non-commutativity (AB ≠ BA in general), associativity A(BC) = (AB)C, and distributive law.
Topic 3
Transpose, Symmetric & Skew-Symmetric Matrices
Transpose properties: (A+B)ᵀ = Aᵀ+Bᵀ, (kA)ᵀ = kAᵀ, (AB)ᵀ = BᵀAᵀ. Symmetric vs skew-symmetric identification. Decomposition: A = ½(A+Aᵀ) + ½(A−Aᵀ).
Topic 4
Elementary Operations & Invertible Matrices
Six elementary transformations (three row, three column). Finding A⁻¹ using elementary row operations on [A|I] → [I|A⁻¹]. Adjoint method: A⁻¹ = (adj A)/|A| for non-singular matrices.
Connections

How this chapter fits in

Useful for setting question difficulty and cross-chapter papers.

Builds on
Class 11 · Sets & Functions
Algebraic structures, equality, and binary operations
Class 10 · Linear Equations
Systems of two equations — extended to matrix form AX = B
Chapter 3 Matrices
Leads to
Ch 4 · Determinants
|A|, cofactors, adj A, Cramer's rule — directly uses matrix concepts
Engineering / Data Science
Linear algebra underpins ML, computer graphics, and signals
Board Blueprint

Marks & question-type breakdown

Typical pattern based on CBSE Class 12 Maths board papers from the last five years.

Question type Marks Typical count What's usually tested
MCQ / Assertion-Reason 1 1–2 Order of a matrix product, identifying symmetric/skew-symmetric, or transpose property
Very Short Answer 2 1 Find a specific element, verify a property, or express as symmetric + skew-symmetric
Short Answer 3 1 Matrix multiplication, proving (AB)ᵀ = BᵀAᵀ, or finding unknown elements using given conditions
Long Answer — Inverse / Equations 5 1 Find A⁻¹ using elementary operations or adjoint method; solve AX = B system of equations
Total (approximate) 8–10 4–5 Weightage varies across paper sets and years
Sample Questions

8 sample questions — generated by MarksZen AI

Aligned to CBSE Class 12 Maths Chapter 3. Covers all question types across Easy, Medium, and Hard difficulty.

Q1 Easy 1 mark MCQ
If A is a matrix of order 3×4 and B is a matrix of order 4×2, then the order of the matrix AB is: (a) 3×2 (b) 4×4 (c) 3×4 (d) 2×3
Q2 Easy 2 marks Short Answer
For the matrix A = [[2, 3], [5, 7]], verify that A + Aᵀ is a symmetric matrix.
Q3 Medium 2 marks Short Answer
If A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 3]], find 2A − 3B.
Q4 Medium 3 marks Short Answer
If A = [[3, 1], [−1, 2]] and B = [[2, −1], [1, 0]], compute AB and BA. Hence verify that AB ≠ BA.
Q5 Medium 3 marks Short Answer
Express the matrix A = [[6, 2, −4], [−3, 5, 1], [8, −1, 0]] as the sum of a symmetric and a skew-symmetric matrix.
Q6 Hard 4 marks Short Answer
Using elementary row operations, find the inverse of: A = [[1, 2, 3], [0, 1, 4], [0, 0, 1]]
Q7 Hard 5 marks Word Problem
Two factories A and B produce three types of products P₁, P₂, P₃. Factory A produces 40 units of P₁, 20 units of P₂, and 10 units of P₃. Factory B produces 30 units of P₁, 50 units of P₂, and 20 units of P₃. The selling prices are ₹100, ₹150, and ₹200 per unit respectively. (i) Represent the production data as a 2×3 matrix and prices as a column matrix. (ii) Use matrix multiplication to find the total revenue of each factory. (iii) Which factory earns more revenue and by how much?
Q8 Hard 5 marks Long Answer
Find the inverse of A = [[2, 1, 3], [5, 2, 1], [3, 1, 2]] using the adjoint method. Hence solve the system of equations: 2x + y + 3z = 9 5x + 2y + z = 3 3x + y + 2z = 7
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Previous Year Questions

From CBSE board examinations

Actual questions from past Class 12 Maths board papers — Matrices chapter.

Board 20222 marks
If A = [[0, 1], [1, 0]], show that A² = I, where I is the identity matrix of order 2. (All India 2022)
Board 20233 marks
If A is a square matrix such that A² = A, show that (I + A)³ = 7A + I. (Delhi 2023)
Board 20205 marks
Using elementary transformations, find the inverse of the matrix A = [[1, 3, −2], [−3, 0, −5], [2, 5, 0]], if it exists. (CBSE 2020)
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FAQ

Questions teachers ask

How many marks does Matrices carry in the CBSE Class 12 board exam? +
Matrices typically carries 8–10 marks in the CBSE Class 12 Mathematics board exam. Questions appear as a 2-mark short answer on matrix operations or transpose properties, a 3-mark question on finding the inverse or solving a matrix equation, and a 5-mark question involving application of matrices to solve a system of linear equations. The exact split varies by paper set and year.
What types of matrices must students know for the CBSE Class 12 board exam? +
Students must know square, rectangular, row, column, zero (null), identity, diagonal, scalar, symmetric, skew-symmetric, and upper/lower triangular matrices. Board questions frequently ask students to express a given matrix as the sum of a symmetric and a skew-symmetric matrix — a standard 3-mark question that appears almost every year.
What is the difference between matrix inverse using elementary operations versus the adjoint method? +
The adjoint method uses A⁻¹ = (adj A) / |A| and works well for 2×2 and 3×3 matrices. Elementary row/column operations (row-reducing [A | I] to [I | A⁻¹]) are also accepted by CBSE. In board exams both methods earn full marks; students should use whichever they practise more confidently since both require similar steps for 3×3 matrices.
Is matrix multiplication commutative? What do board questions test on this? +
No — matrix multiplication is generally NOT commutative: AB ≠ BA in most cases. Board questions often give two matrices and ask students to verify AB ≠ BA, compute both products and compare. They also test the associative property (A(BC) = (AB)C) and distributive property. A common trap: commutativity CAN hold for special cases such as a matrix and its inverse (AA⁻¹ = I) or a matrix and the identity matrix.
How do I generate a custom question paper for Matrices using MarksZen? +
Sign up for a free MarksZen account, choose CBSE Class 12 Mathematics, select Chapter 3 (Matrices), set your preferred question-type mix (MCQ, short answer, long answer) and total marks — the AI generates a complete board-aligned paper with answer key in under 2 minutes, ready for PDF export.