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Sunday, 25 October

JEE Main · Maths · Class 12

Matrices: JEE Main previous year questions

75 questions in 67 of the 108 JEE Main shifts Lumi analysed, asked in 7 of 8 years. Here is what this chapter tests and how.

Study this chapter
Questions
75
in 67 of 108 shifts · 7 of 8 years
Average a shift
0.5
of 25 Maths · 0.6 since 2024 0.1
Typical range
0.3–0.8
questions a shift; fewest 0.0, most 0.8
Weighted estimate
0.6
model, recent years count more · about 2.4 marks
  • Typical range (fact)
  • Average a shift (fact)
  • Weighted estimate (model)

Questions are counted per shift, out of the 25 each subject has, in the shifts Lumi analysed; papers that offered 30 a subject (2017, 2021–2024) are scaled to 25. Typical range: the middle six of the eight years (2017, 2020–2026), leaving out the single highest and lowest year. The average and the range are historical fact. The weighted estimate is a model: each year's share, with recent years counting more (half-life three years), and chapters NTA dropped from the syllabus set near zero. It is a guide to effort, not a forecast of the paper.

What to do

How to spend time on Matrices

Priority

Lower priority

One of the 7 Maths chapters that shared the last fifth of Maths questions in the 108 shifts analysed. How bands are set

Past questions
75in 67 of 108 shifts
Study plan gives it
13 hover 12 months; 10 h over 6
Study this chapter →

Ideas asked most often

  1. Powers of Square Matrices21×
  2. Cayley-Hamilton Theorem Application10×
  3. Binomial Expansion for Commuting Matrices3×
  4. Solving Matrix Equations3×
  5. Row-by-Column Multiplication3×

27 of the 52 ideas in Lumi’s map of this chapter have not been asked or touched in the shifts analysed. Of its idea-tagged questions from 2024 on, 17% tested an idea not asked in an earlier year, so learn the whole chapter, not just the list above.

Within each subject, chapters in NTA's 2026 syllabus are ranked by their average questions a shift over all 8 years analysed (history, not the model). Going down that list, the chapters that together held the first half of the subject's questions are Must do; the next ones, up to 80%, are Should do; the rest are Lower priority.

Key findings

Matrices at a glance

  1. 01

    83%

    of its 18 questions from 2024 on that Lumi matched to an idea tested one JEE Main had asked in an earlier year; 100% came from a topic asked in an earlier year.

  2. 02

    52%

    of the 52 ideas in Lumi's map of the chapter have not been asked or touched on in the shifts analysed.

  3. 03

    40%

    of its questions asked for a numerical answer, 1.3 times the Maths average.

  4. 04

    29%

    of its questions were rated hard, against 26% across all Maths questions in the same shifts.

Questions a shift, year by year

Out of the 25 Maths questions in a shift. Years differ in how many shifts Lumi analysed, so each year is scaled to one shift.

0240’170.5’200.8’210.7’220.6’230.3’240.8’250.7’26

Raw count: 2017: 0 in 1 shift · 2020: 6 in 12 shifts · 2021: 22 in 24 shifts · 2022: 20 in 24 shifts · 2023: 8 in 12 shifts · 2024: 7 in 18 shifts · 2025: 6 in 8 shifts · 2026: 6 in 9 shifts.

Kind of question

SHARE OF QUESTIONS
  • Numerical answer40%
  • Multi-step35%
  • Application11%
  • Calculation8%
  • Conceptual4%
  • Two statements3%

Most common asks: evaluate nested adjoint and inverse of matrix (2); parameters for infinitely many solutions (1); count n for which matrix power equals matrix (1).

Profile

This chapter in numbers

Where it ranks, how steady it is, and what it costs to study.

Rank, all JEE chapters
#63
of 70 by weighted estimate
Rank in Maths
#19
of 21 chapters
Share of Maths
2.1%
of its questions, all shifts analysed
Busiest year
2021
0.8 questions a shift
Year to year
Some swing
varies by about 0.2 questions a shift
Long-run direction
+0.06
questions a shift, per year
Easy questions
12%
29% rated hard
Numerical answer
40%
1.3× the Maths rate
With a figure
0%
graph, diagram or structure
Study time
13 h
10 h on a 6-month plan
Marks per study hour
0.18
#69 of 70 chapters
Foundation for others
94%
of its ideas are used in other chapters

Difficulty, year by year

Each question rated easy, medium or hard. Each bar is that year’s split, so years compare directly; the number is the questions in that year’s analysed shifts (1 to 24 shifts a year).

  • 2017
  • 20206
  • 202122
  • 202220
  • 20238
  • 20247
  • 20256
  • 20266
  • Easy
  • Medium
  • Hard

What it asks you to do

SHARE OF QUESTIONS
  • Multi-step67%
  • Calculation16%
  • Application11%
  • Conceptual7%

Coverage

How much of the chapter JEE Main has used

We split the chapter into 52 ideas. In the 108 shifts analysed, JEE Main built a question around 19 of them. Most of the chapter has never been the main idea of a question, so past questions alone leave most of it out.

Asked as the main idea
19
Touched on in a question
6
Never asked
27

Topics

What repeats, topic by topic

This chapter's questions per topic in each year's analysed shifts. The number is the count; the shade is scaled to one shift, so a year with more analysed shifts (1 to 24 a year) does not look darker. 4 questions had no close topic in Lumi's map and count for the chapter only.

’17’20’21’22’23’24’25’26
015+questions in that year’s analysed shifts, shaded per shift
  • Matrix Multiplication

    As a topic: 60 questions in 7 of 8 years, most often single correct, 40% numerical answer.

  • Matrix Inverse

    As a topic: 51 questions in 8 of 8 years, most often single correct, 25% numerical answer. Recent asks: determinant involving adjoint matrix; numerical: matrix identity with adjoints; singular matrix parameter then extrema of integral.

  • Cramer's Rule

    As a topic: 87 questions in 8 of 8 years, most often single correct, 14% numerical answer. Recent asks: parameters for infinitely many solutions; non-trivial solution then sum trig roots; parameters for no solution of linear system.

  • Permutations and Combinations

    As a topic: 137 questions in 8 of 8 years, most often numerical value, 52% numerical answer. Recent asks: distribute objects with no empty box; count words with letter-type restrictions; count distributions of people to floors.

Drill down

Topic, idea, question

Open a topic to see every idea JEE Main tested inside it, then open an idea to read the questions. Each row shows its 8 years.

  • Open51 · 68%415156533

    18 ideas inside

    • OpenRepeated20 · 39%26542
    • OpenRepeated9 · 18%322
    • OpenRepeated3 · 6%2
    • OpenRepeated2 · 4%
    • OpenRepeated2 · 4%
    • OpenRepeated2 · 4%
    • OpenRepeated2 · 4%
    • Open1 · 2%
    • Open1 · 2%
    • Open1 · 2%
    • Open1 · 2%
    • Open1 · 2%
    • Open1 · 2%
    • Open1 · 2%
    • Open1 · 2%
    • Open1 · 2%
    • Open1 · 2%
    • Open1 · 2%
  • Open14 · 19%3422
  • Open3 · 4%2
  • Open2 · 3%
  • Open1 · 1%

Squares are the 8 years with analysed shifts, 2017 to 2026. A number is the questions that year when there is more than one; the shade is scaled to one shift, so a year with more analysed shifts (1 to 24 a year) does not look darker. Share is of the level above.

Ideas

The exact ideas JEE Main has asked

Inside the topics, 28 different ideas were tested; 9 came more than once. These are the repeats, then the ideas new since 2024.

Dots are the 8 years with analysed shifts, 2017 to 2026.

Every question

All 75 Matrices questions

Newest first, shift by shift. The newest 40 show their opening words; older ones name the idea they tested. Each opens with its options and the official answer.

JEE Main 20266 questions

  1. 2 Apr 2026, Shift 1 · Q4Let A=[121α]A=\begin{bmatrix}1&2\\1&\alpha\end{bmatrix} and B=[33β2]B=\begin{bmatrix}3&3\\\beta&2\end{bmatrix}. If A2−4A+I=OA^2-4A+I=O and B2−5B−6I=OB^2-5B-6I=O,…HardOther
  2. 2 Apr 2026, Shift 2 · Q3If the system of equations x+5y+6z=4,x + 5y + 6z = 4, 2x+3y+4z=7,2x + 3y + 4z = 7, x+6y+az=bx + 6y + az = b has infinitely many solutions, then the point (a,b)(a, b)…MediumSingle correct
  3. 2 Apr 2026, Shift 2 · Q22Consider the matrices A=[2−24−2]A = \begin{bmatrix} 2 & -2 \\ 4 & -2 \end{bmatrix} and B=[−3913]B = \begin{bmatrix} -3 & 9 \\ 1 & 3 \end{bmatrix}. If…MediumNumerical value
  4. 4 Apr 2026, Shift 1 · Q5Let S={A=[abcd]:a,b,c,d∈{0,1,2,3,4} and A2−4A+3I=0}S=\left\{A=\begin{bmatrix}a&b\\c&d\end{bmatrix}: a,b,c,d\in\{0,1,2,3,4\}\text{ and }A^2-4A+3I=0\right\} be a set of 2×22\times2…HardSingle correct
  5. 5 Apr 2026, Shift 2 · Q4Let M be a 3×33\times 3 matrix such that M(100)=(123)M\begin{pmatrix}1\\0\\0\end{pmatrix}=\begin{pmatrix}1\\2\\3\end{pmatrix},…MediumSingle correct
  6. 6 Apr 2026, Shift 2 · Q5Let A=[100310931]A=\begin{bmatrix}1&0&0\\3&1&0\\9&3&1\end{bmatrix} and B=[bij]B=[b_{ij}], 1≤i,j≤31\le i,j\le 3. If B=A99−IB=A^{99}-I, then the value of…MediumSingle correct

JEE Main 20256 questions

  1. 2 Apr 2025, Shift 1 · Q18Let A=[α−16β]A=\begin{bmatrix}\alpha&-1\\6&\beta\end{bmatrix}, α>0\alpha>0, such that det⁡(A)=0\det(A)=0 and α+β=1\alpha+\beta=1. If I denotes 2×22\times2…MediumSingle correct
  2. 2 Apr 2025, Shift 2 · Q3Let AA be a 3×33\times3 real matrix such that A2(A−2I)−4(A−I)=OA^2(A-2I) - 4(A-I) = O, where II and OO are the identity and null matrices,…MediumSingle correct
  3. 4 Apr 2025, Shift 2 · Q4Let the matrix A=[100101010]A=\begin{bmatrix}1&0&0\\1&0&1\\0&1&0\end{bmatrix} satisfy An=An−2+A2−IA^n=A^{n-2}+A^2-I for n≥3n\ge 3. Then the sum of all the…HardSingle correct
  4. 7 Apr 2025, Shift 1 · Q5Let AA be a 3×33 \times 3 matrix such that ∣adj(adj(adj A))∣=81|\mathrm{adj}(\mathrm{adj}(\mathrm{adj}\,A))| = 81. If…HardSingle correct
  5. 7 Apr 2025, Shift 1 · Q22The number of singular matrices of order 2, whose elements are from the set {2,3,6,9}\{2, 3, 6, 9\}, is ___________.MediumNumerical value
  6. 8 Apr 2025, Shift 2 · Q5Let α\alpha be a solution of x2+x+1=0x^2 + x + 1 = 0, and for some aa and bb in R\mathbb{R},…MediumSingle correct

JEE Main 20247 questions

  1. 4 Apr 2024, Shift 2 · Q3Let A=[1201]A=\begin{bmatrix}1&2\\0&1\end{bmatrix} and B=I+adj(A)+(adj A)2+…+(adj A)10B=I+\mathrm{adj}(A)+(\mathrm{adj}\,A)^2+\ldots+(\mathrm{adj}\,A)^{10}. Then, the sum of…MediumSingle correct
  2. 8 Apr 2024, Shift 1 · Q4Let A=[2a013105b]A=\begin{bmatrix}2&a&0\\1&3&1\\0&5&b\end{bmatrix}. If A3=4A2−A−21IA^3=4A^2-A-21I, where II is the identity matrix of order 3×33\times3, then…MediumSingle correct
  3. 8 Apr 2024, Shift 1 · Q22Let A=[2−111]A=\begin{bmatrix}2&-1\\1&1\end{bmatrix}. If the sum of the diagonal elements of A13A^{13} is 3n3^n, then nn is equal to ____________.MediumNumerical value
  4. 9 Apr 2024, Shift 2 · Q4Let B=[1315]B=\begin{bmatrix}1&3\\1&5\end{bmatrix} and AA be a 2×22\times2 matrix such that AB−1=A−1AB^{-1}=A^{-1}. If BCB−1=ABCB^{-1}=A and…HardSingle correct
  5. 1 Feb 2024, Shift 2 · Q21Let A=I2−2MMTA=I_2-2MM^T, where MM is a real matrix of order 2×12\times1 such that the relation MTM=I1M^TM=I_1 holds. If λ\lambda is a real number…HardNumerical value
  6. 27 Jan 2024, Shift 1 · Q4Consider the matrix f(x)=[cos⁡x−sin⁡x0sin⁡xcos⁡x0001]f(x)=\begin{bmatrix}\cos x & -\sin x & 0\\ \sin x & \cos x & 0\\ 0 & 0 & 1\end{bmatrix}. Given below are two…EasyTwo statements
  7. 29 Jan 2024, Shift 1 · Q4Let A be a square matrix such that AAT=IAA^T=I. Then 12A[(A+AT)2+(A−AT)2]\frac{1}{2}A\left[(A+A^T)^2+(A-A^T)^2\right] is equal toMediumSingle correct

JEE Main 20238 questions

  1. 6 Apr 2023, Shift 1 · Q3Let A=[aij]2×2A=[a_{ij}]_{2\times2}, where aij≠0a_{ij}\neq0 for all i, j and A2=IA^2=I. Let aa be the sum of all diagonal elements of AA and b=∣A∣b=|A|.…MediumSingle correct
  2. 6 Apr 2023, Shift 2 · Q4Let PP be a square matrix such that P2=I−PP^2=I-P. For α,β,γ,δ∈N\alpha,\beta,\gamma,\delta\in\mathbb{N}, if Pα+Pβ=γI−29PP^\alpha+P^\beta=\gamma I-29P and…HardSingle correct
  3. 8 Apr 2023, Shift 1 · Q5Let P=[3212−1232]P=\begin{bmatrix}\frac{\sqrt{3}}{2}&\frac{1}{2}\\-\frac{1}{2}&\frac{\sqrt{3}}{2}\end{bmatrix},…HardSingle correct
  4. 8 Apr 2023, Shift 2 · Q4If A=[15λ10]A = \begin{bmatrix} 1 & 5 \\ \lambda & 10 \end{bmatrix}, A−1=αA+βIA^{-1} = \alpha A + \beta I and α+β=−2\alpha + \beta = -2, then…MediumSingle correct
  5. 11 Apr 2023, Shift 1 · Q6Let AA be a 2×22 \times 2 matrix with real entries such that A′=αA+IA' = \alpha A + I, where α∈R−{−1,1}\alpha \in \mathbb{R} - \{-1, 1\}. If…MediumSingle correct
  6. 11 Apr 2023, Shift 1 · Q29Let A=[012a031c0]A = \begin{bmatrix} 0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0 \end{bmatrix}, where a,c∈Ra, c \in \mathbb{R}. If A3=AA^3 = A and the positive…HardNumerical value
  7. 12 Apr 2023, Shift 1 · Q3Let A=[115101]A=\begin{bmatrix}1&\frac{1}{51}\\0&1\end{bmatrix}. If…HardSingle correct
  8. 13 Apr 2023, Shift 1 · Q5The number of symmetric matrices of order 3, with all the entries from the set {0,1,2,3,4,5,6,7,8,9}\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}, isEasySingle correct

JEE Main 202220 questions

  1. 25 Jul 2022, Shift 1 · Q21Let A=(2−1−110−11−10)A=\begin{pmatrix}2&-1&-1\\1&0&-1\\1&-1&0\end{pmatrix} and B=A−IB=A-I. If ω=3 i−12\omega=\frac{\sqrt3\,i-1}{2}, then the number of elements in…HardNumerical value
  2. 25 Jul 2022, Shift 2 · Q23Let A=[1aa01b001]A=\begin{bmatrix}1&a&a\\0&1&b\\0&0&1\end{bmatrix}, a,b∈Ra,b\in\mathbb{R}. If for some n∈Nn\in\mathbb{N},…MediumNumerical value
  3. 26 Jul 2022, Shift 1 · Q4Let A be a 2×22 \times 2 matrix with det⁡ (A)=−1\det\,(\mathrm{A}) = -1 and…HardSingle correct
  4. 26 Jul 2022, Shift 2 · Q3Let A=[111]A=\begin{bmatrix}1\\1\\1\end{bmatrix} and B=[92−102112122132−142−152162172]B=\begin{bmatrix}9^2&-10^2&11^2\\12^2&13^2&-14^2\\-15^2&16^2&17^2\end{bmatrix}, then the…EasySingle correct
  5. 26 Jul 2022, Shift 2 · Q30The number of matrices A=(abcd)A=\begin{pmatrix}a&b\\c&d\end{pmatrix}, where a,b,c,d∈{−1,0,1,2,3,…..,10}a,b,c,d\in\{-1,0,1,2,3,\ldots..,10\}, such that A=A−1A=A^{-1}, is…HardNumerical value
  6. 27 Jul 2022, Shift 1 · Q4Let A=(12−2−5)A = \begin{pmatrix} 1 & 2 \\ -2 & -5 \end{pmatrix}. Let α,β∈R\alpha, \beta \in \mathbb{R} be such that αA2+βA=2I\alpha A^2 + \beta A = 2I.…MediumSingle correct
  7. 27 Jul 2022, Shift 2 · Q4Let A=(4−2αβ)A = \begin{pmatrix} 4 & -2 \\ \alpha & \beta \end{pmatrix}. If A2+γA+18I=OA^2 + \gamma A + 18I = O, then det (A) is equal to __________.MediumSingle correct
  8. 28 Jul 2022, Shift 1 · Q25Let A=[1−12α]A=\begin{bmatrix}1&-1\\2&\alpha\end{bmatrix} and B=[β110], α,β∈RB=\begin{bmatrix}\beta&1\\1&0\end{bmatrix},\ \alpha,\beta\in\mathbf{R}. Let…MediumNumerical value
  9. 28 Jul 2022, Shift 2 · Q63Let A and B be any two 3×33\times 3 symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?MediumSingle correct
  10. 29 Jul 2022, Shift 1 · Q26The number of matrices of order 3×33 \times 3, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is…MediumNumerical value
  11. 29 Jul 2022, Shift 2 · Q2Which of the following matrices can NOT be obtained from the matrix [−121−1]\begin{bmatrix}-1 & 2\\ 1 & -1\end{bmatrix} by a single elementary…EasySingle correct
  12. 29 Jul 2022, Shift 2 · Q23Let X=[111]X=\begin{bmatrix}1\\1\\1\end{bmatrix} and A=[−12301600−1]A=\begin{bmatrix}-1 & 2 & 3\\ 0 & 1 & 6\\ 0 & 0 & -1\end{bmatrix}. For k∈Nk\in\mathbb{N},…MediumNumerical value
  13. 30 Jul 2022, Shift 2 · Q64Let AA be a matrix and…MediumSingle correct
  14. 24 Jun 2022, Shift 2 · Q22Powers of Square Matrices · Hard · Numerical value
  15. 25 Jun 2022, Shift 1 · Q11Powers of Skew-Symmetric Matrices · Medium · Single correct
  16. 25 Jun 2022, Shift 2 · Q21Idempotent and Nilpotent Matrices · Medium · Numerical value
  17. 26 Jun 2022, Shift 2 · Q24Inverse of (I+A) for Nilpotent Matrices · Hard · Numerical value
  18. 28 Jun 2022, Shift 2 · Q27Periodic Matrices · Hard · Numerical value
  19. 29 Jun 2022, Shift 1 · Q10Powers of Square Matrices · Medium · Single correct
  20. 29 Jun 2022, Shift 2 · Q29Square of Skew-Symmetric Matrix · Medium · Numerical value

JEE Main 202122 questions

  1. 3 Aug 2021, Shift 1 · Q62Powers of Square Matrices · Medium · Single correct
  2. 4 Aug 2021, Shift 1 · Q82Row-by-Column Multiplication · Easy · Numerical value
  3. 4 Aug 2021, Shift 2 · Q83Powers of Square Matrices · Medium · Numerical value
  4. 26 Aug 2021, Shift 1 · Q63Properties of Matrix Inverses · Medium · Single correct
  5. 26 Aug 2021, Shift 2 · Q64Powers of Square Matrices · Medium · Single correct
  6. 27 Aug 2021, Shift 1 · Q63Cayley-Hamilton Theorem Application · Medium · Single correct
  7. 31 Aug 2021, Shift 2 · Q81Binomial Expansion for Commuting Matrices · Medium · Numerical value
  8. 24 Feb 2021, Shift 1 · Q83Inverse via Adjoint Formula · Hard · Numerical value
  9. 24 Feb 2021, Shift 1 · Q84Open the question · Hard · Numerical value
  10. 24 Feb 2021, Shift 2 · Q67Skew-Symmetric Coefficient Matrices · Hard · Single correct
  11. 25 Feb 2021, Shift 1 · Q81Inverse of Skew-Symmetric Matrices · Hard · Numerical value
  12. 25 Feb 2021, Shift 2 · Q61Orthogonal Matrix Multiplication Definition · Easy · Single correct
  13. 26 Feb 2021, Shift 1 · Q73Trace of A times A Transpose · Medium · Single correct
  14. 26 Feb 2021, Shift 2 · Q86Cayley-Hamilton Theorem Application · Hard · Numerical value
  15. 22 Jul 2021, Shift 2 · Q64Powers of Square Matrices · Medium · Single correct
  16. 22 Jul 2021, Shift 2 · Q82Non-commutativity Property · Hard · Numerical value
  17. 25 Jul 2021, Shift 1 · Q87Powers of Square Matrices · Medium · Numerical value
  18. 25 Jul 2021, Shift 2 · Q75Powers of Square Matrices · Easy · Single correct
  19. 16 Mar 2021, Shift 2 · Q88Row-by-Column Multiplication · Medium · Numerical value
  20. 17 Mar 2021, Shift 2 · Q90Eigenvalue-Type Homogeneous Systems · Medium · Numerical value
  21. 18 Mar 2021, Shift 1 · Q63Open the question · Easy · Single correct
  22. 18 Mar 2021, Shift 2 · Q82Cayley-Hamilton Theorem Application · Medium · Numerical value

JEE Main 20206 questions

  1. 8 Jan 2020, Shift 1 · Q72Trace of A times A Transpose · Medium · Numerical value
  2. 8 Jan 2020, Shift 2 · Q54Inverse via Characteristic Equation · Easy · Single correct
  3. 2 Sep 2020, Shift 1 · Q54Open the question · Medium · Two statements
  4. 2 Sep 2020, Shift 2 · Q54Orthogonal Matrix Properties · Medium · Single correct
  5. 3 Sep 2020, Shift 1 · Q72Powers of Square Matrices · Medium · Numerical value
  6. 4 Sep 2020, Shift 1 · Q54Powers of Square Matrices · Medium · Single correct

Questions from the official JEE Main papers published by NTA. Chapter, topic and idea tags, counts and notes are Lumi’s. Pattern: Asked most years.

More Maths, Class 12

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