Maths · Class 12 · Matrices and Determinants

Matrices and Determinants for JEE

Everything you need for this topic on one page: the ideas, the formulas, the traps, and free lectures that explain it well.

Last updated: 26 October 2026

Direct answer

Matrix algebra, special matrices, determinants and their properties, adjoint and inverse, and solving linear systems.

Key ideas

Matrices and Determinants: what you must understand

AB ≠ BA in general
|kA| = kⁿ|A| for an n×n matrix
A⁻¹ exists only if |A| ≠ 0
System AX = B: |A| ≠ 0 gives a unique solution

Formula sheet

Must-know formulas for Matrices and Determinants

A⁻¹ = adj(A)/|A|
|adj A| = |A|^(n−1)
|AB| = |A||B|, (AB)⁻¹ = B⁻¹A⁻¹
Area of triangle = ½|det[x y 1]|

One-page formula sheet

Printable Matrices and Determinants formula sheet: formulas, key ideas and traps on one page. Save it as a PDF for offline revision.

Avoid

Common mistakes students make

Using |kA| = k|A|
Wrong order in (AB)ᵀ = BᵀAᵀ
Missing the infinite-solution vs no-solution case

For the exam

How this topic is asked in JEE

Appears in almost every JEE Main shift. Almost every shift. Properties of adjoint and determinant give quick marks.

NCERT map

Where Matrices and Determinants is in NCERT

Chapter names follow the current rationalised NCERT textbooks. Read these sections before solving PYQs.

Class	Book	Chapter	Sections JEE asks from
Class 12	Mathematics Part I	Matrices	Types of Matrices; Operations on Matrices; Symmetric and Skew Symmetric Matrices; Invertible Matrices
Class 12	Mathematics Part I	Determinants	Properties of Determinants; Area of a Triangle; Adjoint and Inverse of a Matrix; Applications of Determinants and Matrices

JEE Advanced level

A JEE Advanced-level example on Matrices and Determinants

Question. Let A be a 3×3 matrix satisfying A² = 3A - 2I, where I is the identity matrix. If det(A) = 4, find the value of det(A³ - 2A² + A).

Show solution

From A² = 3A - 2I, compute A³ = A·A² = A(3A - 2I) = 3A² - 2A = 3(3A - 2I) - 2A = 7A - 6I
Substitute into B = A³ - 2A² + A: B = (7A - 6I) - 2(3A - 2I) + A = 7A - 6I - 6A + 4I + A = 2A - 2I = 2(A - I)
So det(B) = det(2(A - I)) = 2³ × det(A - I) = 8 × det(A - I)
From A² - 3A + 2I = 0, eigenvalues satisfy λ² - 3λ + 2 = 0, giving λ = 1 or λ = 2
Since det(A) = 4 = product of eigenvalues, the eigenvalues must be 1, 2, 2. So eigenvalues of (A - I) are 0, 1, 1
det(A - I) = 0 × 1 × 1 = 0, therefore det(B) = 8 × 0 = 0

Answer: 0

This example was solved twice independently and both solutions agree.

Free lectures

Best free lectures for Matrices and Determinants

Full-chapter free lectures picked from YouTube search results by views, from established JEE channels (mostly in Hinglish). Use the subtopic list to jump straight to the part you need (timestamps come from each video's own chapter list). Watch one, then solve PYQs the same day.

MATRICES in One Shot: All Concepts & PYQs Covered | JEE Main & Advanced — JEE Wallah, 6:23:26
Jump to a subtopic (30)
- 00:00 Introduction
- 07:57 Topics to be covered
- 09:28 Matrix Definition
- 15:38 Order of a Matrix
- 20:02 Questions
- 31:04 Types of Matrices (With Examples)
- 01:08:03 Brain Teaser 1
- 01:14:10 Algebra of Matrices
- 01:40:25 Questions
- 02:05:44 Properties of Matrix Multiplication
- 02:19:22 Trace of a Matrix (Questions)
- 02:34:10 Brain Teaser 2
- 02:40:50 The Transpose of a Matrix & Its Properties
- 02:54:57 Symmetric and Skew Symmetric Matrices
- 03:07:08 Properties of Symmetric and Skew Symmetric Matrices
- 03:26:00 Break
- 03:47:43 Mid Lecture Recap
- 03:51:16 Adjoint of a Matrix
- 04:07:05 Properties of the Adjoint of a Matrix
- 04:28:40 Questions & PYQ's
- 04:47:18 Inverse of a Matrix
- 04:54:58 Properties of the Inverse of a Matrix
- 05:05:04 Characteristics Equation
- 05:09:00 Cayley Hamilton Theorem
- 05:11:30 Questions
- 05:30:56 Types of Square Matrices
- 05:43:30 Questions
- 05:52:20 Matrix Method for Solving Linear Equations
- 05:57:08 Questions
- 06:20:00 Thank you
Matrices And Determinants One Shot | JEE Main 2024 | RRR | Arvind Kalia Sir — JEE Nexus by Unacademy, 2:41:55
Jump to a subtopic (12)
- 0:00 Introduction
- 4:06 Square Matrix
- 12:12 Algebra of Matrices
- 39:06 Transpose & Its properties
- 40:55 Some special Matrices
- 1:03:40 Determinants (Basics, relation b/w determinant & Matrix)
- 1:13:39 Properties of Determinants
- 1:33:42 Differentiation of determinant
- 1:35:51 A special determinant
- 1:38:42 Adjoint & Inverse of Matrices
- 2:08:09 System of linear equations
- 2:27:20 Cayley Hamilton theorem

Practise

Where to practise for free

16 JEE Main 2026 PYQs on Matrices and Determinants, with solutions.
Read the chapter in NCERT textbooks (official, free), then solve its NCERT Exemplar (official, free) questions.
Chapter-wise past questions: JEE Main and Advanced PYQs and question bank.
Timed practice tests: NTA Abhyas practice tests (official, free) and our free mock tests.
See it move: interactive simulation.

FAQs

Frequently asked questions

Is Matrices and Determinants important for JEE Main 2027?

Appears in almost every JEE Main shift. It is part of the official 'Matrices and Determinants' unit in the JEE Main 2027 syllabus.

What are the most important formulas in Matrices and Determinants?

A⁻¹ = adj(A)/|A|; |adj A| = |A|^(n−1); |AB| = |A||B|, (AB)⁻¹ = B⁻¹A⁻¹.

How should I study Matrices and Determinants for JEE?

Understand the key ideas from NCERT or one full lecture, learn the formula sheet, then solve 25–30 past JEE questions on the topic. Almost every shift. Properties of adjoint and determinant give quick marks.