📐 CBSE · Class 12 · Mathematics · Chapter 4

Chapter 4
Determinants

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

5Topics
8–10Board marks
8Sample questions
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Key Formulas — Chapter 4
  • 2×2 det: |A| = ad − bc for A = [[a,b],[c,d]]
  • Cofactor: A_ij = (−1)^(i+j) × M_ij
  • Inverse: A⁻¹ = (1/|A|) × adj(A), |A| ≠ 0
  • Cramer's rule: x = D₁/D, y = D₂/D, z = D₃/D
  • Area of triangle: ½ |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|
  • Singular matrix: |A| = 0 ⟹ A⁻¹ does not exist
Chapter Overview

What this chapter covers

A determinant is a scalar value computed from the elements of a square matrix. Chapter 4 of NCERT Class 12 Mathematics introduces determinants of 2×2 and 3×3 matrices, their evaluation by cofactor expansion along any row or column, and the eight fundamental properties of determinants — tools that allow students to simplify complex determinants without brute-force arithmetic.

The chapter extends to minors, cofactors, the adjoint, and the inverse of a matrix. The key relationship A⁻¹ = (1/|A|) × adj(A) ties determinants directly to matrix algebra. Students learn to classify a matrix as singular (|A| = 0, no inverse exists) or non-singular (|A| ≠ 0, inverse exists), and to use the determinant to find the area of a triangle given three coordinate vertices.

The chapter culminates in solving a system of linear equations using matrix methods — expressing Ax = B as x = A⁻¹B — and via Cramer's rule, where each unknown is the ratio of two determinants. Board questions consistently test the 5-mark long answer on solving a 3×3 system, making this the highest-stakes application in the chapter.

Topics

What's inside Chapter 4

As per NCERT Class 12 Mathematics (CBSE syllabus)

Topic 1
Determinants of Matrices (Order 1, 2, 3)
Definition and evaluation of determinants for 1×1, 2×2, and 3×3 matrices. Expansion along any row or column using cofactors. Recognising that the value is the same regardless of the row or column chosen.
Topic 2
Properties of Determinants
Eight properties: row/column interchange reverses sign; scalar factor extraction; zero row/column gives zero determinant; identical rows/columns give zero; row operations preserve value; sum property; and det(AB) = det(A)·det(B). Used to simplify before evaluating.
Topic 3
Area of a Triangle & Collinearity
Area = ½|Δ| where Δ is the 3×3 determinant formed from the coordinates of the three vertices. Points are collinear if and only if this determinant equals zero — a direct board application of the sign-less modulus condition.
Topic 4
Minors, Cofactors, Adjoint & Inverse
Minor M_ij is the (n−1)×(n−1) sub-determinant; cofactor A_ij = (−1)^(i+j)M_ij. Adjoint = transpose of cofactor matrix. Inverse: A⁻¹ = adj(A)/|A| when |A| ≠ 0. Singular vs non-singular classification.
Topic 5
Solving Linear Equations — Matrix & Cramer's Methods
Matrix method: express system as Ax = B, then x = A⁻¹B. Cramer's rule: x = D₁/D, y = D₂/D, z = D₃/D. Conditions for unique solution (D ≠ 0), no solution, or infinitely many solutions (D = 0 with inconsistency checks).
Connections

How this chapter fits in

Useful for setting question difficulty and cross-chapter papers.

Builds on
Ch 3 · Matrices
Matrix notation, order, operations — prerequisite for cofactor and adjoint concepts
Class 10 · Linear Equations
Two-variable systems solved by substitution/elimination — generalized here to 3 variables
Chapter 4 Deter-
minants
Leads to
Ch 6 · Application of Derivatives
Jacobian determinants and maxima/minima problems in higher studies
Class 12 · Vectors & 3D Geometry
Cross product of vectors expressed as a 3×3 determinant; scalar triple product
Board Blueprint

Marks & question-type breakdown

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

Question type Marks Typical count What's usually tested
MCQ / Assertion-Reason 1 1–2 Value of a 2×2 or 3×3 determinant, singular matrix condition, or property identification
Very Short Answer 2 1 Find cofactor / adjoint of a 2×2 matrix; area of triangle using determinant
Short Answer 3 1 Evaluate a 3×3 determinant using properties; find the value of k for a singular matrix
Long Answer — Matrix/Cramer's 5 1 Solve a system of 3 linear equations using matrix inverse method or Cramer's rule
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 4. Covers all question types across Easy, Medium, and Hard difficulty.

Q1 Easy 1 mark MCQ
If A = [[3, 2], [1, 4]], the value of |A| is: (a) 10 (b) 14 (c) 5 (d) −10
Q2 Easy 2 marks Short Answer
Find the value of x if the matrix A = [[x, 3], [2, 6]] is singular.
Q3 Medium 2 marks Short Answer
Using the determinant formula, find the area of the triangle with vertices A(1, 0), B(6, 0), and C(4, 3).
Q4 Medium 3 marks Short Answer
Evaluate the following determinant using properties of determinants: | 1 2 3 | | 3 5 9 | | 2 4 6 |
Q5 Medium 3 marks Short Answer
Find the adjoint of the matrix A = [[1, 2], [3, 4]] and verify that A · adj(A) = |A| · I.
Q6 Hard 4 marks Short Answer
If A = [[2, 3, 1], [1, −1, 2], [3, 1, −1]], find A⁻¹ using the adjoint method. Show all cofactors clearly.
Q7 Hard 5 marks Long Answer
Solve the following system of linear equations using the matrix (inverse) method: x + 2y − 3z = −4 2x + 3y + 2z = 2 3x − 3y − 4z = 11
Q8 Hard 5 marks Long Answer
Using Cramer's rule, solve the system: 2x − y + z = 3 x + 2y − z = −1 −x + y + 2z = 4 Also state the condition under which Cramer's rule is applicable.
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Previous Year Questions

From CBSE board examinations

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

Board 20222 marks
If A is a square matrix of order 3 with |A| = 4, find the value of |2A|. (CBSE All India 2022)
Board 20235 marks
Using matrices, solve the following system of linear equations: 2x + 3y + 3z = 5, x − 2y + z = −4, 3x − y − 2z = 3. (CBSE Delhi 2023)
Board 20203 marks
Using properties of determinants, show that: | a+b b+c c+a | | b+c c+a a+b | = −2(a³ + b³ + c³ − 3abc). (CBSE 2020) | c+a a+b b+c |
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FAQ

Questions teachers ask

How many marks does Determinants carry in the CBSE Class 12 Maths board exam? +
Determinants typically carries 8–10 marks in the CBSE Class 12 Mathematics board exam. Questions appear as one 1-mark MCQ on properties or singular matrices, one 2-mark short answer (adjoint or cofactors), and one 5-mark long answer involving the matrix-inverse method or Cramer's rule for a system of equations. This pattern has held across most CBSE sets for the last five years.
What are the most important properties of determinants for board exams? +
The six properties tested most often in CBSE board papers are: (1) interchanging two rows/columns reverses the sign, (2) a common factor of any row/column can be taken outside, (3) if two rows or columns are identical the determinant is zero, (4) adding a scalar multiple of one row to another leaves the determinant unchanged, (5) the determinant of a triangular matrix equals the product of diagonal entries, and (6) det(AB) = det(A) × det(B). Properties (1), (3), and (4) are the core tools for evaluation problems.
What is the difference between a minor and a cofactor in CBSE Class 12? +
The minor M_ij of element a_ij is the determinant of the (n−1)×(n−1) sub-matrix obtained by deleting the i-th row and j-th column. The cofactor A_ij = (−1)^(i+j) × M_ij — it is the signed minor. Board questions often ask students to find the cofactor matrix or use cofactors to expand the determinant along a specific row or column. Remembering the sign pattern (+−+) / (−+−) / (+−+) saves time.
How is the inverse of a matrix related to its determinant and adjoint? +
For an invertible (non-singular) matrix A, A⁻¹ = (1/|A|) × adj(A), where adj(A) is the transpose of the cofactor matrix of A. This formula is directly tested in 5-mark board questions that ask students to find the inverse and then use it to solve a system of three linear equations. A matrix is invertible if and only if |A| ≠ 0; when |A| = 0 the matrix is singular and has no inverse.
How do I generate a custom question paper for Determinants using MarksZen? +
Sign up for a free MarksZen account, choose CBSE Class 12 Mathematics, select Chapter 4 (Determinants), 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.