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 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 S be the set of all column matrices b 1 b 2 b 3 such that b 1 , b 2 , b 3 ∈ ℝ and the system of equations (in real variables) - x + 2 y + 5 z = b 1 2 x - 4 y + 3 z = b 2 x - 2 y + 2 z = b 3 has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution of each b 1 b 2 b 3 ∈ S ?

x + 2 y + 5 z = b 1 , 2 x + 3 z = b 2 and x + 4 y - 5 z = b 3

x + y + 3 z = b 1 , 5 x + 2 y + 6 z = b 2 and - 2 x - y - 3 z = b 3

– x + 2 y - 5 z = b 1 , 2 x - 4 y + 10 z = b 2 and x - 2 y + 5 z = b 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

MEDIUM

JEE Advanced

IMPORTANT

Earn 100

For $3 \times 3$ matrices $M$ and $N$ , which of the following statement(s) is (are) NOT correct?

14.29% 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 - 2019>Q 2

Let

P_{1} = I = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}], P_{2} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}], P_{3} = [\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}],

P_{4} = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix}], P_{5} = [\begin{matrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}], P_{6} = [\begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix}],

and

X = \sum_{k = 1}^{6} P_{K} [\begin{matrix} 2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 2 & 1 \end{matrix}] P_{K}^{T}

where

P_{K}^{T}

denotes the transpose of the matrix

P_{K} .

Then which of the following options is/are correct?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 7 - Matrices and Determinants>JEE Advanced Paper 2 - 2019>Q 3

Let

x \in R

and let

P = [\begin{matrix} 1 & 1 & 1 \\ 0 & 2 & 2 \\ 0 & 0 & 3 \end{matrix}], Q = [\begin{matrix} 2 & x & x \\ 0 & 4 & 0 \\ x & x & 6 \end{matrix}]

and

R = P Q P^{- 1} .

Then which of the following options is/are correct?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 7 - Matrices and Determinants>JEE Advanced Paper 2 - 2018>Q 4

Let

S

be the set of all column matrices

[\begin{matrix} b_{1} \\ b_{2} \\ b_{3} \end{matrix}]

such that

b_{1}, b_{2}, b_{3} \in ℝ

and the system of equations (in real variables)

\begin{matrix} - x + 2 y + 5 z = b_{1} \\ 2 x - 4 y + 3 z = b_{2} \\ x - 2 y + 2 z = b_{3} \end{matrix}

has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution of each

[\begin{matrix} b_{1} \\ b_{2} \\ b_{3} \end{matrix}] \in S

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 7 - Matrices and Determinants>JEE Advanced Paper 2 - 2018>Q 5

Let P be a matrix of order

3 \times 3

such that all the entries in P are from the set

\{- 1, 0, 1\}

. Then, the maximum possible value of the determinant of P is _______.

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 7 - Matrices and Determinants>JEE Advanced Paper 2 - 2017>Q 6

The number of

3 \times 3

matrices

M

with entries from

{0,1, 2}

, such that the sum of the diagonal elements of

M^{T} M

5

, are

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 7 - Matrices and Determinants>JEE Advanced Paper 2 - 2016>Q 7

Let

P = [\begin{matrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{matrix}]

and

I

be the identity matrix of order

3

. If

Q = [q_{i j}]

is a matrix such that

P^{50} - Q = I,

then the value of

\frac{q_{31} + q_{32}}{q_{21}}

is equal to

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 7 - Matrices and Determinants>JEE Advanced Paper 2 - 2016>Q 8

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 ?

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 =

For 3×3 matrices M and N, which of the following statement(s) is (are) NOT correct?

Important Questions on Matrices and Determinants

Learn and Prepare for any exam you want !

For $3 \times 3$ matrices $M$ and $N$ , which of the following statement(s) is (are) NOT correct?