Core Syllabus Module

Matrices and Determinants

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Chapter Elite-20 Challenge Zone

Target Advanced

JEE Advanced 2024

LEVELJEE Advanced

Let $α$ and $β$ be the distinct roots of the equation $x^{2} + x - 1 = 0$ . Consider the set $T = {1, α, β}$ . For a $3 \times 3$ matrix $M = (a_{ij})_{3 \times 3}$ , define $R_{i} = a_{i 1} + a_{i 2} + a_{i 3}$ and $C_{j} = a_{1 j} + a_{2 j} + a_{3 j}$ for $i = 1, 2, 3$ and $j = 1, 2, 3$ . Match each entry in List-I to the correct entries in List-II.

List-I

(P)

The number of matrices

M = (a_{ij})_{3 \times 3}

with all entries in

T

such that

R_{i} = C_{j} = 0

for all

i, j

, is

(Q)

The number of symmetric matrices

M = (a_{ij})_{3 \times 3}

with all entries in

T

such that

C_{j} = 0

for all

j

, is

(R)

Let

M = (a_{ij})_{3 \times 3}

be a skew symmetric matrix such that

a_{ij} \in T

for

i > j

. Then the number of elements in the set

⎩ ⎨ ⎧ x y z : x, y, z \in R, M x y z = a_{12} 0 - a_{23} ⎭ ⎬ ⎫

(S)

Let

M = (a_{ij})_{3 \times 3}

be a matrix with all entries in

T

such that

R_{i} = 0

for all

i

. Then the absolute value of determinant of

M

List-II

Let $p, q, r$ be nonzero real numbers that are, respectively, the $1 0^{t h}, 10 0^{t h}$ and $100 0^{t h}$ terms of a harmonic progression. Consider the system of linear equations $x + y + z = 1$ $10 x + 100 y + 1000 z = 0$ $q r x + p r y + pq z = 0$ .

List-I

(P)

q / r = 10

, then the system of linear equations has

(Q)

p / r \neq = 100

, then the system of linear equations has

(R)

p / q \neq = 10

, then the system of linear equations has

(S)

p / q = 10

, then the system of linear equations has

List-II

(1)

x = 0, y = 10/9, z = - 1/9

as a solution

(2)

x = 10/9, y = - 1/9, z = 0

as a solution

(3)

infinitely many solutions

(4)

no solution

(5)

at least one solution

JEE Main 2020 - 8 Jan (Evening)

LEVELJEE Advanced

The system of linear equations $λ x + 2 y + 2 z = 5 2 x + 3 y + 5 z = 8 4 x + λ y + 6 z = 10$ has

(A)

Infinite solutions when

λ = 8

(B)

Infinite solutions when

λ = 2

(C)

no solutions when

λ = 8

(D)

no solutions when

λ = 2

JEE Advanced 2010

LEVELJEE Advanced

Comprehension Passage

Let

p

be an odd prime number and

T_{p}

be the following set of

2 \times 2

matrices :

T_{p} = {A = [a c b a] : a, b, c \in {0, 1, 2, \dots, p - 1}}

Question 1:

The number of $A$ in $T_{p}$ such that $A$ is either symmetric or skew-symmetric or both, and $det (A)$ divisible by $p$ is

(A)

(p - 1)^{2}

(B)

2 (p - 1)

(C)

(p - 1)^{2} + 1

(D)

2 p - 1

Question 2:

The number of $A$ in $T_{p}$ such that the trace of $A$ is not divisible by $p$ but $det (A)$ is divisible by $p$ is

(A)

(p - 1) (p^{2} - p + 1)

(B)

p^{3} - (p - 1)^{2}

(C)

(p - 1)^{2}

(D)

(p - 1) (p - 2)

Question 3:

The number of $A$ in $T_{p}$ such that $det (A)$ is not divisible by $p$ is

(A)

2 p^{2}

(B)

p^{3} - 5 p

(C)

p^{3} - 3 p

(D)

p^{3} - p^{2}

JEE(ADVANCED)-201

LEVELJEE Advanced

How many $3 \times 3$ matrices M with entries from ${0, 1, 2}$ are there, for which the sum of the diagonal entries of $M^{T} M$ is 5?

JEE Main 2022 (27 June Shift 2)

LEVELJEE Advanced

Let for some real numbers $α$ and $β$ , $a = α - i β$ . If the system of equations $4 i x + (1 + i) y = 0$ and $8 (cos \frac{2 π}{3} + i sin \frac{2 π}{3}) x + \overset{a}{ˉ} y = 0$ has more than one solution then $\frac{α}{β}$ is equal to :

(A)

- 2 + 3

(B)

2 - 3

(C)

2 + 3

(D)

- 2 - 3

JEE Main 2020 - 2 Sep (Evening)

LEVELJEE Advanced

Let $A = {X = (x, y, z)^{T} : P X = 0 and x^{2} + y^{2} + z^{2} = 1}$ , where $P = 1 - 2 1 239 1 - 4 - 1$ then the set $A$ :

(A)

is an empty set.

(B)

contains more than two elements.

(C)

contains exactly two elements.

(D)

is a singleton.

JEE Main 2026 (22 January Shift 1)

LEVELJEE Advanced

Let A be a $3 \times 3$ matrix such that $A + A^{T} = O$ . If $A 1 - 1 0 = 332, A^{2} 1 - 1 0 = - 3 19 - 24$ and $det (adj (2 adj (A + I))) = (2)^{α} \cdot (3)^{β} \cdot (11)^{γ}, α, β, γ$ are non-negative integers, then $α + β + γ$ is equal to .........

JEE Main 2022 (25 July Shift 1)

LEVELJEE Advanced

The number of $θ \in (0, 4 π)$ for which the system of linear equations $3 (sin 3 θ) x - y + z = 2$ , $3 (cos 2 θ) x + 4 y + 3 z = 3$ , $6 x + 7 y + 7 z = 9$ has no solution is :

JEE Main 2023 (31 January Shift 1)

LEVELJEE Advanced

For the system of linear equations $x + y + z = 6$ , $α x + β y + 7 z = 3$ , $x + 2 y + 3 z = 14$ , which of the following is NOT true ?

(A)

α = β = 7

, then the system has no solution

(B)

α = β

and

α \neq = 7

then the system has a unique solution.

(C)

There is a unique point

(α, β)

on the line

x + 2 y + 18 = 0

for which the system has infinitely many solutions

(D)

For every point

(α, β) \neq = (7, 7)

on the line

x - 2 y + 7 = 0

, the system has infinitely many solutions.

JEE Main 2023 (13 April Shift 2)

LEVELJEE Advanced

Let $A = 1 α 1 231312$ for $α \in R$ . If $∣ A ∣ = 2$ . If $∣2 adj (2 adj (2 A)) ∣ = 3 2^{n}$ , then $3 n + α$ is equal to

Comprehension Passage

Let

a, b

and

c

be three real numbers satisfying

[a b c] 187923777 = [000] \dots (E)

Question 1:

If the point $P (a, b, c)$ , with reference to (E), lies on the plane $2 x + y + z = 1$ , then the value of $7 a + b + c$ is

Let $ω$ be a solution of $x^{3} - 1 = 0$ with $Im (ω) > 0$ , if $a = 2$ with $b$ and $c$ satisfying (E), then the value of $\frac{3}{ω ^{a}} + \frac{1}{ω ^{b}} + \frac{3}{ω ^{c}}$ is equal to

Let $b = 6$ , with $a$ and $c$ satisfying (E). If $α$ and $β$ are the roots of the quadratic equation $a x^{2} + b x + c = 0$ , then $\sum_{n = 0}^{\infty} (\frac{1}{α} + \frac{1}{β})^{n}$ is

\infty

JEE Main 2020 - 6 Sep (Evening)

LEVELJEE Advanced

Let $θ = \frac{π}{5}$ and $A = (cos θ - sin θ sin θ cos θ)$ . If $B = A + A^{4}$ then $d e t (B)$ :

Comprehension Passage

Let

A

be the set of all

3 \times 3

symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.

Question 1:

The number of matrices in $A$ is

The number of matrices $A$ in $A$ for which the system of linear equations $A x y z = 100$ has a unique solution, is

(A)

less than 4

(B)

at least 4 but less than 7

(C)

at least 7 but less than 10

(D)

at least 10

Question 3:

The number of matrices $A$ in $A$ for which the system of linear equations $A x y z = 100$ is inconsistent, is

Let $R = ⎩ ⎨ ⎧ a c 0 325 b d 0 : a, b, c, d \in {0, 3, 5, 7, 11, 13, 17, 19} ⎭ ⎬ ⎫$ . Then the number of invertible matrices in $R$ is

JEE Advanced 2007

LEVELJEE Advanced

Consider the following linear equations $a x + b y + cz = 0; b x + cy + a z = 0; c x + a y + b z = 0$ Match the conditions/expressions in Column I with statements in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.

List-I

(P)

a + b + c \neq = 0

and

a^{2} + b^{2} + c^{2} = ab + b c + c a

(Q)

a + b + c = 0

and

a^{2} + b^{2} + c^{2} \neq = ab + b c + c a

(R)

a + b + c \neq = 0

and

a^{2} + b^{2} + c^{2} \neq = ab + b c + c a

(S)

a + b + c = 0

and

a^{2} + b^{2} + c^{2} = ab + b c + c a

List-II

(1)

the equations represent planes meeting only at a single point

(2)

the equations represent the line

x = y = z

(3)

the equations represent identical planes.

(4)

the equations represent the whole of the three dimensional space.

JEE Advanced 2008

LEVELJEE Advanced

Match the Statements/Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.

List-I

(P)

The minimum value of

\frac{x ^{2} + 2 x + 4}{x + 2}

(Q)

Let A and B be

3 \times 3

matrices of real numbers, where A is symmetric, B is skew-symmetric, and

(A + B) (A - B) = (A - B) (A + B)

. If

(A B)^{t} = (- 1)^{k} A B

, where

(A B)^{t}

is the transpose of the matrix AB, then the possible values of

k

are

(R)

Let

a = lo g_{3} lo g_{3} 2

. An integer

k

satisfying

1 < 2^{- k + 3^{- a}} < 2

, must be less than

(S)

sin θ = cos ϕ

, then the possible values of

\frac{1}{π} (θ \pm ϕ - \frac{π}{2})

are

List-II

If $A = a 11 1 b b 0 d c, B = a 0 f 1 d g 1 c h, U = f g h, V = a^{2} 00, X = x y z$ and $A X = U$ has infinitely many solutions, prove that $B X = V$ has no unique solution. Also show that if $a f d \neq = 0$ , then $B X = V$ has no solution.

JEE Main 2022 (24 June Shift 2)

LEVELJEE Advanced

Let the system of linear equations $x + y + α z = 2$ , $3 x + y + z = 4$ , $x + 2 z = 1$ have a unique solution $(x^{}, y^{}, z^{})$ . If $(α, x^{})$ , $(y^{}, α)$ and $(x^{}, - y^{*})$ are collinear points, then the sum of absolute values of all possible values of $α$ is :

JEE Main 2025 (January)

LEVELJEE Advanced

Let M denote the set of all real matrices of order $3 \times 3$ and let $S = {- 3, - 2, - 1, 1, 2}$ . Let $S_{1} = {A = [a_{ij}] \in M : A = A^{T} and a_{ij} \in S, \forall i, j}$ $S_{2} = {A = [a_{ij}] \in M : A = - A^{T} and a_{ij} \in S, \forall i, j}$ $S_{3} = {A = [a_{ij}] \in M : a_{11} + a_{22} + a_{33} = 0 and a_{ij} \in S, \forall i, j}$ If $n (S_{1} \cup S_{2} \cup S_{3}) = 125 α$ then $α$ equals

Micro-Topic Strategy Matrix

Types of Matrices

Algebraic Operations on Matrices

Properties of Determinants

Minors and Cofactors

Adjoint and Inverse of a Matrix

Solution of System of Linear Equations (Matrix Method and Cramer's Rule)

Chapter Elite-20 Challenge Zone

List-I

List-II

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The system of linear equations λx+2y+2z=52x+3y+5z=84x+λy+6z=10 has

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of A in Tp​ such that A is either symmetric or skew-symmetric or both, and det(A) divisible by p is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of A in Tp​ such that the trace of A is not divisible by p but det(A) is divisible by p is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of A in Tp​ such that det(A) is not divisible by p is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } How many 3×3 matrices M with entries from {0,1,2} are there, for which the sum of the diagonal entries of MTM is 5?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A={X=(x,y,z)T:PX=0 and x2+y2+z2=1}, where P=​1−21​239​1−4−1​​ then the set A :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of θ∈(0,4π) for which the system of linear equations 3(sin3θ)x−y+z=2, 3(cos2θ)x+4y+3z=3, 6x+7y+7z=9 has no solution is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For the system of linear equations x+y+z=6, αx+βy+7z=3, x+2y+3z=14, which of the following is NOT true ?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A=​1α1​231​312​​ for α∈R. If ∣A∣=2. If ∣2 adj(2 adj(2A))∣=32n, then 3n+α is equal to

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the point P(a,b,c), with reference to (E), lies on the plane 2x+y+z=1, then the value of 7a+b+c is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let ω be a solution of x3−1=0 with Im(ω)>0, if a=2 with b and c satisfying (E), then the value of ωa3​+ωb1​+ωc3​ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let b=6, with a and c satisfying (E). If α and β are the roots of the quadratic equation ax2+bx+c=0, then ∑n=0∞​(α1​+β1​)n is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let θ=5π​ and A=(cosθ−sinθ​sinθcosθ​). If B=A+A4 then det(B) :

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of matrices in A is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of matrices A in A for which the system of linear equations A​xyz​​=​100​​ has a unique solution, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of matrices A in A for which the system of linear equations A​xyz​​=​100​​ is inconsistent, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let R=⎩⎨⎧​​ac0​325​bd0​​:a,b,c,d∈{0,3,5,7,11,13,17,19}⎭⎬⎫​. Then the number of invertible matrices in R is

List-I

List-II

List-I

List-II

The system of linear equations $λ x + 2 y + 2 z = 5 2 x + 3 y + 5 z = 8 4 x + λ y + 6 z = 10$ has

The number of $A$ in $T_{p}$ such that $A$ is either symmetric or skew-symmetric or both, and $det (A)$ divisible by $p$ is

The number of $A$ in $T_{p}$ such that the trace of $A$ is not divisible by $p$ but $det (A)$ is divisible by $p$ is

The number of $A$ in $T_{p}$ such that $det (A)$ is not divisible by $p$ is

How many $3 \times 3$ matrices M with entries from ${0, 1, 2}$ are there, for which the sum of the diagonal entries of $M^{T} M$ is 5?

Let $A = {X = (x, y, z)^{T} : P X = 0 and x^{2} + y^{2} + z^{2} = 1}$ , where $P = 1 - 2 1 239 1 - 4 - 1$ then the set $A$ :

The number of $θ \in (0, 4 π)$ for which the system of linear equations $3 (sin 3 θ) x - y + z = 2$ , $3 (cos 2 θ) x + 4 y + 3 z = 3$ , $6 x + 7 y + 7 z = 9$ has no solution is :

For the system of linear equations $x + y + z = 6$ , $α x + β y + 7 z = 3$ , $x + 2 y + 3 z = 14$ , which of the following is NOT true ?

Let $A = 1 α 1 231312$ for $α \in R$ . If $∣ A ∣ = 2$ . If $∣2 adj (2 adj (2 A)) ∣ = 3 2^{n}$ , then $3 n + α$ is equal to

If the point $P (a, b, c)$ , with reference to (E), lies on the plane $2 x + y + z = 1$ , then the value of $7 a + b + c$ is

Let $ω$ be a solution of $x^{3} - 1 = 0$ with $Im (ω) > 0$ , if $a = 2$ with $b$ and $c$ satisfying (E), then the value of $\frac{3}{ω ^{a}} + \frac{1}{ω ^{b}} + \frac{3}{ω ^{c}}$ is equal to

Let $b = 6$ , with $a$ and $c$ satisfying (E). If $α$ and $β$ are the roots of the quadratic equation $a x^{2} + b x + c = 0$ , then $\sum_{n = 0}^{\infty} (\frac{1}{α} + \frac{1}{β})^{n}$ is

Let $θ = \frac{π}{5}$ and $A = (cos θ - sin θ sin θ cos θ)$ . If $B = A + A^{4}$ then $d e t (B)$ :

The number of matrices in $A$ is

The number of matrices $A$ in $A$ for which the system of linear equations $A x y z = 100$ has a unique solution, is

The number of matrices $A$ in $A$ for which the system of linear equations $A x y z = 100$ is inconsistent, is

Let $R = ⎩ ⎨ ⎧ a c 0 325 b d 0 : a, b, c, d \in {0, 3, 5, 7, 11, 13, 17, 19} ⎭ ⎬ ⎫$ . Then the number of invertible matrices in $R$ is