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JEE Advanced Matrices and Determinants questions with solutions

Matrix algebra, determinants, inverse and linear systems. 31 questions from our 2016–2025 JEE Advanced pattern papers, from the Class 12 Mathematics syllabus. Each has the answer and a full solution.

Also practise: Matrices and Determinants for JEE Main

JEE Advanced 2025

Let A be a real 2× 2 matrix that is not a scalar multiple of I and satisfies A^2-3A+2I=O. Then det (A^6-3A^-1) equalsJEE Advanced 2025 Paper 1 · medium
Let N=beginpmatrix0&2&1\\0&0&2\\0&0&0endpmatrix and A=I+N. If ak denotes the element in the first row and third column of A^k, then Σk=1^10ak equalsJEE Advanced 2025 Paper 2 · hard

JEE Advanced 2024

Let A=beginpmatrix2&1&0\\0&2&1\\0&0&2endpmatrix. Which of the following statements is/are correct?JEE Advanced 2024 Paper 1 · hard
Then det (A^100-100A) equalsJEE Advanced 2024 Paper 2 · easy

JEE Advanced 2023

Let a,b,c be the roots of x^3-3x^2+2x-1=0 and let A=beginpmatrixa&b&c\c&a&b\b&c&aendpmatrix. Then det (adj(A) A^T equalsJEE Advanced 2023 Paper 1 · medium
Consider the system of linear equations in x,y,z: x+y+z=6, x+2y+3z=14, x+2y+az=b, where a,b are real constants.JEE Advanced 2023 Paper 2 · medium
Let A be a 3×3 real matrix with A^2=A+2I and tr(A)=3. The value of tr(A^5+det (adjA) isJEE Advanced 2023 Paper 2 · medium
Let M=beginpmatrix1&2\\0&1endpmatrix and let N=M^50-I. The sum of all entries of N isJEE Advanced 2023 Paper 2 · easy

JEE Advanced 2022

Let A=beginpmatrix2&1&0\\0&3&1\\0&0&1endpmatrix and suppose A^5=pA^2+qA+rI for real numbers p,q,r. Find p-q+r.JEE Advanced 2022 Paper 1 · hard
List-I (P) det (adj(adjA)) (Q) det (2A^-1) (R) trace of adjA (S) det (A^TA^2 List-II (1) 3 (2) 4 (3) 6 (4) 8 (5) 16JEE Advanced 2022 Paper 1 · medium
Let mathcal S be the set of all 3×3 matrices A with every entry in \0,1\ such that A^2=I, where I is the identity matrix. The number of elements of…JEE Advanced 2022 Paper 2 · hard
Let A=beginpmatrix1&2\\2&1endpmatrix and let I be the 2×2 identity matrix.JEE Advanced 2022 Paper 2 · medium

JEE Advanced 2021

Paragraph 3: Let A be a 3×3 real matrix whose characteristic polynomial is λ^3-6λ^2+11λ-5. Denote its eigenvalues (counted with multiplicity) by…JEE Advanced 2021 Paper 1 · 2-question set
Let M=beginpmatrix2&-1\\1&0endpmatrix and let n be a positive integer.JEE Advanced 2021 Paper 1 · medium
Let M=beginpmatrix1&2&0\\0&1&2\\0&0&1endpmatrix and let I be the identity matrix of order 3.JEE Advanced 2021 Paper 2 · medium
Paragraph 1: Define sequences (xn) and (yn) by beginpmatrixxn\ynendpmatrix=A^nbeginpmatrix1\\0endpmatrix for n≥0. It is known that the characteristic…JEE Advanced 2021 Paper 2 · 2-question set

JEE Advanced 2020

Let P=beginpmatrix1&1&0\\0&1&1\\0&0&1endpmatrix and Q=[qij]=P^40+P^-40. The value of q13+q22 isJEE Advanced 2020 Paper 1 · medium
Find the number of 2×2 matrices A with every entry in \-1,0,1\ such that A^3=I, where I is the identity matrix.JEE Advanced 2020 Paper 2 · hard
Let M=beginpmatrix1&k&0\\0&1&k\k&0&1endpmatrix where k is a real number.JEE Advanced 2020 Paper 2 · medium

JEE Advanced 2019

For a positive integer n, the (1,3) entry of M^n+M^-n equals 1444. Then n equalsJEE Advanced 2019 Paper 1 · easy
Let M=beginpmatrix1&2&0\\0&1&2\\0&0&1endpmatrix and let n be a positive integer.JEE Advanced 2019 Paper 2 · medium

JEE Advanced 2018

Let A=beginpmatrix1&1&0\\0&1&1\\0&0&1endpmatrix and let I be the 3×3 identity matrix.JEE Advanced 2018 Paper 1 · medium
Paragraph 1: For integers n≥0 define xn,yn by beginpmatrixxn\ynendpmatrix=M^nbeginpmatrix1\\0endpmatrix, so that (x0,y0)=(1,0), (x1,y1)=(2,1)…JEE Advanced 2018 Paper 1 · 2-question set
Find the sum of all nine entries of M^-15.JEE Advanced 2018 Paper 2 · medium

JEE Advanced 2017

Let M=beginpmatrix2&1&1\\1&2&1\\1&1&2endpmatrix and let adjM denote its adjoint.JEE Advanced 2017 Paper 1 · medium
The sum of all four entries of the matrix A^-50 isJEE Advanced 2017 Paper 2 · medium

JEE Advanced 2016

Let N=beginpmatrix0&1&2\\0&0&3\\0&0&0endpmatrix and M=I+N.JEE Advanced 2016 Paper 1 · medium
Let A=beginpmatrix3&-1\\4&-1endpmatrix and B=A+A^2+…+A^10. The sum of all four entries of B isJEE Advanced 2016 Paper 2 · medium

Other Mathematics topics

Area Under Curves 13 Binomial Theorem 10 Complex Numbers 24 Conic Sections 34 Differential Equations 18 Differentiation and Application of Derivatives 20 Indefinite and Definite Integrals 18 Limits 19 Permutations and Combinations 18 Probability 38 Quadratic Equations 12 Sequences and Series 11 Sets, Relations and Functions 8 Straight Lines and Circles 10 Three Dimensional Geometry 22 Trigonometry and Inverse Trigonometric Functions 18 Vector Algebra 9