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

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

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

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

32.26% 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 - 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 =

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?

Let S be the set of all column matrices b1b2b3 such that b1, b2, b3∈ℝ and the system of equations (in real variables) -x+2y+5z=b12x-4y+3z=b2x-2y+2z=b3 has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution of each b1b2b3∈ S ?

Important Questions on Matrices and Determinants

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