The JEE Advanced is one of India's hardest engineering entrance examinations, which focuses on testing conceptual knowledge and solving problems. Matrices and Determinants are an important chapter in the mathematics syllabus for JEE Advanced because they are also used in algebra, calculus, and coordinate geometry.

Based upon past year questions, the topic Matrices and Determinants made up an average of around 8% -10% of the total marks in mathematics, although weightage may vary every year as per examination pattern. In mathematics, typically 2-3 questions are drafted on this topic in JEE Advanced, where the total marks are in the range of about 8-12 overall.

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Here are some of the important JEE Advanced Previous Year Questions on the topic Matrices and Determinants.

Question 1:

Let \( S = \left\{ \begin{pmatrix} 0 & 1 & c \\ 1 & a & d \\ 1 & b & e \end{pmatrix} : a, b, c, d, e \in \{0, 1\} \text{ and } |A| \in \{-1, 1\} \right\} \), where \( |A| \) denotes the determinant of A. Then the number of elements in \( S \) is ______.

[JEE Advanced - 2024]
View Solution

Question 2:

Let \( \alpha \) and \( \beta \) be the distinct roots of the equation \( x^2 + x - 1 = 0 \). Consider the set \( T = \{1, \alpha, \beta\} \). For a 3 × 3 matrix \( M = (a_{ij})_{3 \times 3} \), define \( R_i = a_{11} + a_{12} + a_{13} \) and \( C_j = a_{11} + a_{21} + a_{31} \) for \( i = 1, 2, 3 \) and \( j = 1, 2, 3 \). Match each entry in List-I to the correct entry in List-II.

[JEE Advanced - 2024]

A) (P) → (4) (Q) → (2) (R) → (5) (S) → (1)

B) (P) → (2) (Q) → (4) (R) → (1) (S) → (5)

C) (P) → (2) (Q) → (4) (R) → (3) (S) → (5)

D) (P) → (1) (Q) → (5) (R) → (3) (S) → (4)

View Solution

Question 3:

Let \( \beta \) be a real number. Consider the matrix

\( A = \begin{pmatrix} \beta & 0 & 1 \\ 2 & 1 & -2 \\ 3 & 1 & -2 \end{pmatrix} \)

If \( A^7 - (\beta - 1) A^6 - \beta A^5 \) is a singular matrix, then the value of \( 9\beta \) is ______.

[JEE Advanced - 2023]

A) 1

B) 2

C) 3

D) 4

View Solution

Question 4:

Let \( M = (a_{ij}), i, j \in \{1, 2, 3\} \), be the 3×3 matrix such that \( a_{ij} = 1 \) if \( j + 1 \) is divisible by \( i \), otherwise \( a_{ij} = 0 \). Then which of the following statements is(are) true?

[JEE Advanced - 2023]

A) \( M \) is invertible

B) There exists a nonzero column matrix \( \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} \) such that \( M \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} = \begin{pmatrix} a_1 \\ -a_2 \\ -a_3 \end{pmatrix} \)

C) The set \( \{ x \in \mathbb{R}^3 : Mx = 0 \} \neq 0 \), where \( 0 = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} \)

D) The matrix \( (M - 2I) \) is invertible, where \( I \) is the 3×3 identity matrix

View Solution

Question 5:

If \( M = \begin{pmatrix} 5 & 2 & 3 \\ 3 & 2 & 1 \\ -2 & 1 & 2 \end{pmatrix} \), then which of the following matrices is equal to \( M^{2022} \)?

[JEE Advanced - 2022]

A) \( \begin{pmatrix} 3034 & 3033 \\ -3033 & -3032 \end{pmatrix} \)

B) \( \begin{pmatrix} 3034 & 3033 \\ -3033 & -3032 \end{pmatrix} \)

C) \( \begin{pmatrix} 3034 & -3033 \\ 3033 & -3032 \end{pmatrix} \)

D) \( \begin{pmatrix} 3033 & 3032 \\ -3032 & -3031 \end{pmatrix} \)

View Solution

Question 6:

Let \( \hat{i}, \hat{j}, \hat{k} \) be the unit vectors along the three positive coordinate axes. Let

\( \vec{a} = 3\hat{i} + \hat{j} - \hat{k}, \quad \vec{b} = \hat{i} + b_2 \hat{j} + b_3 \hat{k}, \quad \vec{c} = c_1 \hat{i} + c_2 \hat{j} + c_3 \hat{k} \)

be three vectors such that \( b_2b_3 > 0 \), \( \vec{a} \cdot \vec{b} = 0 \), and the matrix equation holds:

\( \begin{pmatrix} 0 & -c_3 & c_2 \\ c_3 & 0 & -c_1 \\ -c_2 & c_1 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ b_2 \\ b_3 \end{pmatrix} = \begin{pmatrix} 3 \\ 1 - c_2 \\ -1 - c_3 \end{pmatrix} \)

Then, which of the following is/are TRUE?

[JEE Advanced - 2022]

A) \( \vec{a} \cdot \vec{c} = 0 \)

B) \( \vec{b} \cdot \vec{c} = 0 \)

C) \( |\vec{b}| > \sqrt{10} \)

D) \( |\vec{c}| \leq \sqrt{11} \)

View Solution

Question 7:

For any \( 3 \times 3 \) matrix \( M \), let \( |M| \) denote the determinant of \( M \). Let

\( E = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 8 & 13 & 18 \end{pmatrix}, \quad P = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \quad F = \begin{pmatrix} 1 & 3 & 2 \\ 8 & 18 & 13 \\ 2 & 4 & 3 \end{pmatrix} \)

If \( Q \) is a non-singular matrix of order \( 3 \times 3 \), then which of the following statements is/are TRUE?

[JEE Advanced - 2021]

A) \( F = PEP \) and \( P^2 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \)

B) \( |EQ + PFQ^{-1}| = |EQ| + |PFQ^{-1}| \)

C) \( |(EF)^3| > |EF|^2 \)

D) Sum of the diagonal entries of \( P^{-1} EP + F \) is equal to the sum of diagonal entries of \( E + P^{-1} FP \)

View Solution

Question 8. Comprehension:

Let \( \alpha, \beta \) and \( \gamma \) be real numbers such that the system of linear equations

\( x + 2y + 3z = \alpha \)
\( 4x + 5y + 6z = \beta \)
\( 7x + 8y + 9z = \gamma - 1 \)

is consistent. Let \( |M| \) represent the determinant of the matrix.

\( M = \begin{pmatrix} \alpha & 2 & \gamma \\ \beta & 1 & 0 \\ -1 & 0 & 1 \end{pmatrix} \)

Let \( P \) be the plane containing all those \( (\alpha, \beta, \gamma) \) for which the above system of linear equations is consistent, and \( D \) be the square of the distance of the point \( (0, 1, 0) \) from the plane \( P \).

Question 1:

The value of \( |M| \) is ______ ?

[JEE Advanced - 2021]

Question 2:

The value of \( D \) is ______.

[JEE Advanced - 2021]
View Solution

Question 9:

The trace of a square matrix is defined to be the sum of its diagonal entries. If \( A \) is a \( 2 \times 2 \) matrix such that the trace of \( A \) is 3 and the trace of \( A^3 \) is -18, then the value of the determinant of \( A \) is ______.

[JEE Advanced - 2020]

A) 0

B) 1

C) -1

D) 2

View Solution

Question 10:

Let \( x \in \mathbb{R} \) and let

\( P = \begin{pmatrix} 1 & 1 & 1 \\ 0 & 2 & 2 \\ 0 & 0 & 3 \end{pmatrix}, \quad Q = \begin{pmatrix} 2 & x & x \\ 0 & 4 & 0 \\ x & x & 6 \end{pmatrix}, \quad R = PQP^{-1} \)

Then which of the following options is/are correct?

[JEE Advanced - 2019]

A) There exists a real number \( x \) such that \( PQ = QP \)

B) \( \text{det} R = \text{det} \begin{pmatrix} 2 & x & x \\ 0 & 4 & 0 \\ x & x & 5 \end{pmatrix} + 8, \) for all \( x \in \mathbb{R} \)

C) For \( x = 0 \), if \( R \begin{pmatrix} 1 \\ a \\ b \end{pmatrix} = \begin{pmatrix} 6 \\ a \\ b \end{pmatrix} \), then \( a + b = 5 \)

D) For \( x = 1 \), there exists a unit vector \( \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} \) for which \( R \begin{pmatrix} \alpha \\ \beta \\ \gamma \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} \)

View Solution

Question 11:

If \( 2A + 3B = \begin{pmatrix} 2 & -1 & 4 \\ 3 & 2 & 5 \end{pmatrix} \) and \( A + 2B = \begin{pmatrix} 5 & 0 & 3 \\ 1 & 6 & 2 \end{pmatrix} \), then \( B = \) ______.

[JEE Advanced - 2006]

A) \( \begin{pmatrix} 8 & 1 & 2 \\ 1 & 10 & 1 \end{pmatrix} \)

B) \( \begin{pmatrix} 8 & 1 & -2 \\ -1 & 10 & -1 \end{pmatrix} \)

C) \( \begin{pmatrix} 8 & 1 & 2 \\ -1 & 10 & -1 \end{pmatrix} \)

D) \( \begin{pmatrix} 8 & -1 & 2 \\ -1 & 10 & -1 \end{pmatrix} \)

View Solution

JEE Advanced Matrices and Determinants Question Trends (2014-2024)

Year-wise Question Distribution

Here is a year-wise distribution of important topics and their difficulty level.

Year No. of Questions Key Topics Tested Difficulty Level
2024 1 Determinant properties, Matrix equations Moderate
2023 2 Adjoint & Inverse, System of equations Easy-Moderate
2022 1 Eigenvalues, Special matrices Moderate
2021 2 Matrix recurrence relations, Determinant expansion Hard
2020 1 Orthogonal matrices, Rank Moderate
2019 1 Vandermonde determinant, Factor theorem Hard
2018 2 Nilpotent matrices, Cramer’s Rule Moderate-Hard
2017 1 Skew-symmetric determinant, Trig-based Hard
2016 1 Consistency of linear system Moderate
2015 1 Involutory matrices, Matrix algebra Easy
2014 2 Determinant properties, Inverse Moderate

Topic-wise Weightage (2014–2024)

Below is the distribution of topic with their frequency in the JEE Advanced exam in previous years.

Topic Frequency Most Asked Concepts
Determinants 60% Expansion, Factor theorem, Skew-symmetric
Matrix Algebra 30% Inverse, Adjoint, Special matrices (Idempotent/Nilpotent)
System of Equations 10% Cramer’s Rule, Consistency

Difficulty Analysis

Difficulty Percentage (2014–2024) Example Questions
Easy 20% 2015 (Involutory matrix check)
Moderate 50% 2023 (Adjoint calculation)
Hard 30% 2017 (Trigonometric determinant)

Important Theorems/Properties

Here are some of the most important theorems that are asked frequently in JEE Advanced Exam

  • Cayley-Hamilton Theorem
  • Vandermonde Determinant
  • Rank & Consistency

JEE Advanced Mathematics Paper Analysis (2024 & Trends)

The mathematics syllabus of JEE Advanced assesses students in their analytical thinking, problem-solving abilities, and understanding of concepts. Two papers (Paper 1 and Paper 2), last three hours each, with approximately 15 - 18 questions covering the syllabus collectively. The questions are based/inspired by every part of the syllabus with varying levels of abilities and complexity, ranging from theoretical concepts to challenging numerical problems, requiring a mixture of properties and applications from several disciplines.

Overview (2024)

Aspect Details
Total Questions Approximately 15–18 per paper (2 papers)
Difficulty Moderate to Hard; around 30% easy, 40% moderate, and 30% difficult
Question Types Single/Multiple MCQs, Numerical Value, Paragraph-based, Match-the-Column
Key Focus Areas Differential Calculus, Integral Calculus, Matrices & Determinants, Probability, Vector and 3D Geometry

Chapter-Wise Weightage (2024)

Topic Weightage Difficulty Remarks
Differential Calculus 8% Moderate Emphasis on application of derivatives and limits
Integral Calculus 19% Moderate-Hard Focus on definite integration and differential equations
Matrices & Determinants 8% Moderate Application-based questions on matrix operations and properties
Probability 14% Moderate Problems involving complex probability distributions
Vector and 3D Geometry 11% Moderate Questions on vector algebra and three-dimensional geometry concepts

Difficulty Trends (2019–2024)

Year Difficulty Level Key Observations
2024 Moderate-Hard There are an more numerical problems consisting of several concepts
2023 Moderate There was a balanced theory and numericals but lengthy problems in calculus
2022 Hard Focussed on complex algebraic manipulations and advanced calculus
2021 Very Hard It had unusual problems in coordinate geometry and vector algebra
2020 Moderate The concept were more NCERT-aligned and straightforward questions in algebra and trigonometry

Key Takeaways for 2025 Aspirants

  • Concentrate on High-Weightage Topics: It may be beneficial to study topics like Integral Calculus, Probability, and Vector & 3D Geometry as they appear frequently.
  • Time Management: Some questions are long, so consider practicing problems as if they were timed to help with efficiency.
  • Clarity of Concept: Understanding the basic concepts early in the process may be important so that you can apply them effectively to more difficult questions.

JEE Advanced Exam Pattern

The JEE Advanced exam is considered as one of the top engineering entrance examinations in India, which takes place annually for enrollment at IITs and other top institutions to undergraduate programs. JEE Advanced consists of two compulsory papers, Paper 1 and Paper 2, each with three hours of duration. It challenges students’ knowledge of Physics, Chemistry and Mathematics. The exam consists of different question types including Multiple Choice Questions (MCQs), Numerical Value-based Questions and Assertion-Reasoning questions to evaluate the foundational knowledge of topics and ability to solve problems. Both papers have a total of 180 marks, with marks deducted for incorrect answers in a few question types.

Aspect Details
Mode Computer-Based Test (CBT)
Papers 2 (Paper 1 & Paper 2) – Both mandatory
Duration 3 hours per paper (with break between papers)
Total Subjects Physics, Chemistry, Mathematics (Equal weightage)
Total Questions ~54–60 (18–20 per subject, split across question types)
Language English & Hindi (option to switch during exam)

The exam is known for its challenging nature and is conducted in English and Hindi. It is essential to clear the JEE Main exam before appearing for JEE Advanced.

Marking Scheme with Question types in JEE Advanced

Here is a detailed marking scheme for different questions in JEE Advanced exam.

Question Type Marks per Q Negative Marking Examples
Single Correct MCQ +3 -1 Basic theory/direct formula questions
Multiple Correct MCQ +4 (full) -2 (if all wrong) / Partial marks* "Which of the following is/are true?"
Numerical Value (NV) +3 No negative Calculations (e.g., wavelength)
Paragraph-Based Varies (2–4) Depends on sub-type Linked to a common scenario
Match the Following +2 per match -1 per wrong match Column matching (e.g., graphs)

Partial Marks: For Multiple Correct MCQs, +1 per correct option (if not all selected).

Subject-Wise Distribution

Here is the subject wise questions distribution in the JEE Advanced exam describing number of questions in each subject.

Subject MCQs Numerical Paragraph/Matching Total
Physics 6–8 4–5 2–3 12–16
Chemistry 6–8 4–5 2–3 12–16
Maths 6–8 4–5 2–3 12–16

Subject wise Syllabus: JEE Advanced 2025

Here is a quick syllabus of JEE Advanced 2025 subject wise which helps in preparation of the students. Check JEE Advanced Syllabus for a detailed Syllabus.

Subject Key Topics
Physics Mechanics, Electrodynamics, Modern Physics, Optics, Thermodynamics, Error Analysis
Chemistry Physical (Thermo, Kinetics), Organic (Reactions), Inorganic (Coordination, P-Block)
Maths Algebra, Calculus, Coordinate Geometry, Trigonometry, Vectors

Note:

  • Physics: Focus on Modern Physics (20% weightage) & Electrodynamics.
  • Chemistry: Organic reactions & Coordination compounds are high-scoring.
  • Maths: Calculus (35% weightage) is most critical.

In addition to conceptual knowledge, you should practice multiple mock tests before your exam.

Check:

JEE Advanced Mock Test

JEE Advanced Previous Year Question Papers with Detailed Solutions

JEE Advanced 2024 Question Paper Pdf with Solutions

Paper Name Question Paper with Solutions
JEE Advanced 2024 Paper 1 Question Paper (English) Check Solutions
JEE Advanced 2024 Paper 2 Question Paper (English) Check Solutions
JEE Advanced 2024 Paper 1 Question Paper (Hindi) Check Solutions
JEE Advanced 2024 Paper 2 Question Paper (Hindi) Check Solutions
Architecture Aptitude Test Check Solutions

JEE Advanced Question Paper has 4 sections- Section 1 (Maximum marks-12), Section 2 (Maximum marks-12), Section 3 (Maximum marks- 24) and Section 4 (Maximum marks-12)

JEE Advanced 2023 Question Paper PDF Download With Solutions

Paper Answer Key PDF
Question Paper 1 PDF Check Solutions
Question Paper 2 PDF Check Solutions
Question Paper AAT PDF Check Solutions

JEE Advanced 2022 Question Paper

2022 Question Paper PDF Download
Question Paper 1 PDF Click here
Question Paper 2 PDF Click here

JEE Advanced 2021 Question Paper

Question Paper PDF Download
Question Paper 1 PDF Physics Chemistry Maths
Question Paper 2 PDF Physics Chemistry Maths

JEE Advanced 2020 Question Paper

Question Paper PDF Download
Question Paper 1 PDF Click here
Question Paper 2 PDF Click here

Frequently Asked Questions (FAQs)

Q1. Are matrices and determinants important for JEE Advanced?

Answer. Absolutely, matrices and determinants are crucial topics for JEE Advanced. They have enormous weight in the Mathematics section, with concepts such as properties of determinants, matrix operations, and applications appearing consistently in the past years' papers. A solid understanding of these subjects can help you accumulate important marks in the examination.

Q2. Is 95 percentile enough to qualify for JEE advanced?

Answer. Generally, a 95 percentile is not sufficient to qualify for JEE Advanced. The qualifying cutoff for JEE Advanced is usually higher. It can be around 90-95% and in some cases even higher, depending on the difficulty of the exam and the number of candidates who have appeared for it. Therefore, it is always advisable and good to prepare to score higher than the eligibility.

Q3. Is PNC important for JEE Advanced?

Answer. Certainly, Permutation and Combination (PNC) is a crucial topic for JEE Advanced. It has a good weightage in the Mathematics section and can be tested in objective type as well as numerical based questions. A considerable understanding of the concepts of PNC can help solve difficult problems and can improve your score in the exam.

Q4. What is the toughest subject in JEE?

Answer. The hardest subject in JEE differs from person to person. Physics is often considered the toughest exam by many as the hardest subject because it has so many concepts and problem-solving/application-based questions to focus on (especially in Mechanics, Electrodynamics, and Modern Physics). Some students think of Mathematics or Chemistry (especially Organic Chemistry) as the hardest subject, depending on the student's strengths/weaknesses. The bottom line is that the hard subject depends on how each student prepares and understands the material.

Q5. Is matrices and determinants a hard chapter?

Answer. Matrices and Determinants can be tough for some students because it is more of an abstract idea as well as one must know what the properties of the determinants are, what are the applications and operations of the matrices, etc. However, knowing that matrices and determinants are a part of your JEE preparation, with adequate practice and understanding of the basic principles, this chapter is easier. It is an important topic in JEE Advanced, and understanding and mastering it can increase your Mathematics score.Absolutely, matrices and determinants are crucial topics for JEE Advanced. They have enormous weight in the Mathematics section, with concepts such as properties of determinants, matrix operations, and applications appearing consistently in the past years' papers. A solid understanding of these subjects can help you accumulate important marks in the examination.