For any 3 × 3 matrix M , let | M | denote the determinant of M . Let E = 1 2 3 2 3 4 8 13 18 , P = 1 0 0 0 0 1 0 1 0 & F = 1 3 2 8 18 13 2 4 3 If Q is a nonsingular matrix of order 3 × 3 , then which of the following statements is (are) TRUE ?

Sum of the diagonal entries of P - 1 E P + F is equal to the sum of diagonal entries of E + P - 1 F P

Let M = 0 1 a 1 2 3 3 b 1 and a d j M = - 1 1 - 1 8 - 6 2 - 5 3 - 1 where a and b are real numbers. Which of the following options is/are correct?

If M = α β γ = 1 2 3 , then α - β + γ = 3

For any 3 × 3 matrix M , let | M | denote the determinant of M . Let E = 1 2 3 2 3 4 8 13 18 , P = 1 0 0 0 0 1 0 1 0 & F = 1 3 2 8 18 13 2 4 3 If Q is a nonsingular matrix of order 3 × 3 , then which of the following statements is (are) TRUE ?

Sum of the diagonal entries of P - 1 E P + F is equal to the sum of diagonal entries of E + P - 1 F P

Let x ∈ R and let P = 1 1 1 0 2 2 0 0 3 , Q = 2 x x 0 4 0 x x 6 and R = P Q P - 1 . Then which of the following options is/are correct?

For x = 0 , if R 1 a b = 6 1 a b , then a + b = 5

For x = 1 , there exists a unit vector α i ^ + β j ^ + γ k ^ for which R α β γ = 0 0 0

There exists a real number x such that P Q = Q P

Let P 1 = I = 1 0 0 0 1 0 0 0 1 , P 2 = 1 0 0 0 0 1 0 1 0 , P 3 = 0 1 0 1 0 0 0 0 1 , P 4 = 0 1 0 0 0 1 1 0 0 , P 5 = 0 0 1 1 0 0 0 1 0 , P 6 = 0 0 1 0 1 0 1 0 0 , and X = ∑ k = 1 6 P K 2 1 3 1 0 2 3 2 1 P K T where P K T denotes the transpose of the matrix P K . Then which of the following options is/are correct?

X - 30 I is an invertible matrix

Let M = 0 1 a 1 2 3 3 b 1 and a d j M = - 1 1 - 1 8 - 6 2 - 5 3 - 1 where a and b are real numbers. Which of the following options is/are correct?

If M = α β γ = 1 2 3 , then α - β + γ = 3

HARD

JEE Advanced

IMPORTANT

Earn 100

Let $a, λ, μ \in ℝ$ . Consider the system of linear equations

$a x + 2 y = λ$

$3 x - 2 y = μ$

Which of the following statement(s) is (are) correct ?

52.17% studentsanswered this correctly

Important Questions on Matrices and Determinants

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 7 - Matrices and Determinants>JEE Advanced Paper 2 - 2013>Q 9

Let

ω

be a complex cube root of unity with

ω \neq 1

and

P = [p_{i j}]

be a

n \times n

matrix with

p_{i j} = ω^{i + j}

. Then

P^{2} \neq 0

, when

n =

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 7 - Matrices and Determinants>JEE Advanced Paper 1 - 2021>Q 11

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

$E = [\begin{matrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 8 & 13 & 18 \end{matrix}], P = [\begin{array}{l} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}] & F = [\begin{matrix} 1 & 3 & 2 \\ 8 & 18 & 13 \\ 2 & 4 & 3 \end{matrix}]$

If $Q$ is a nonsingular matrix of order $3 \times 3$ , then which of the following statements is (are) TRUE ?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 7 - Matrices and Determinants>JEE Advanced Paper 1 - 2021>Q 12

For any

3 \times 3

matrix

M

, let

| M |

denote the determinant of

M

. Let

I

be the

3 \times 3

identity matrix. Let

E

and

F

be two

3 \times 3

matrices such that

(I - E F)

is invertible. If

G = (I - E F)^{- 1}

, then which of the following statements is (are) TRUE ?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 7 - Matrices and Determinants>JEE Advanced Paper 1 - 2020>Q 13

Let

M

be a

3 \times 3

invertible matrix with real entries and let

I

denote the

3 \times 3

identity matrix.

M^{- 1} = a d j (a d j M)

, then which of the following statements is/are ALWAYS TRUE?

MEDIUM

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 7 - Matrices and Determinants>JEE Advanced Paper 1 - 2019>Q 14

Let

M = [\begin{matrix} \sin^{4} θ & - 1 - \sin^{2} θ \\ 1 + \cos^{2} θ & \cos^{4} θ \end{matrix}] = α I + β M^{- 1},

where

α = α (θ)

and

β = β (θ)

are real number, and

I

is the

2 \times 2

identity matrix. If

α^{*}

is the minimum of the set {

α (θ) : θ \in [0, 2 π)}

and

β^{*}

is the minimum of the set

\{β (θ) : θ \in [0, 2 π)\},

then the value of

α^{*} + β^{*}

MEDIUM

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 7 - Matrices and Determinants>JEE Advanced Paper 1 - 2019>Q 15

Let

M = [\begin{matrix} 0 & 1 & a \\ 1 & 2 & 3 \\ 3 & b & 1 \end{matrix}]

and

a d j M = [\begin{matrix} - 1 & 1 & - 1 \\ 8 & - 6 & 2 \\ - 5 & 3 & - 1 \end{matrix}]

where

a

and

b

are real numbers. Which of the following options is/are correct?

MEDIUM

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 7 - Matrices and Determinants>JEE Advanced Paper 1 - 2019>Q 16

Let

S

be the sample space of all

3 \times 3

matrices with entries from the set

\{0, 1\} .

Let the events

E_{1}

and

E_{2}

be given by

E_{1} = \{A \in S : d e t A = 0\}

and

E_{2} = {A \in S :

sum of entries of

A

7} .

If a matrix is chosen at random from

S,

then the conditional probability

P (E_{1}| E_{2})

equals____

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 7 - Matrices and Determinants>JEE Advanced Paper 1 - 2017>Q 17

Which of the following is (are) NOT the square of a

3 \times 3

matrix with real entries?

Let a, λ, μ ∈ℝ . Consider the system of linear equations ax+2y=λ 3x-2y=μ Which of the following statement(s) is (are) correct ?

Important Questions on Matrices and Determinants

Learn and Prepare for any exam you want !

Let $a, λ, μ \in ℝ$ . Consider the system of linear equations

$a x + 2 y = λ$

$3 x - 2 y = μ$

Which of the following statement(s) is (are) correct ?