For the system of linear equations a x + y + z = 1 , x + a y + z = 1 , x + y + a z = β , which one of the following statements is NOT correct?

It has infinitely many solutions if α = 2 and β = - 1

It has no solution if α = - 2 and β = 1

x + y + z = 3 4 if α = 2 and β = 1

It has infinitely many solutions if α = 1 and β = 1

Let the system of linear equations x + y + k z = 2 2 x + 3 y - z = 1 3 x + 4 y + 2 z = k have infinitely many solutions. Then the system k + 1 x + 2 k - 1 y = 7 2 k + 1 x + k + 5 y = 10 has :

unique solution satisfying x + y = 1

unique solution satisfying x - y = 1

HARD

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IMPORTANT

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Let $M$ be any $3 \times 3$ matrix with entries from the set $\{0, 1, 2\}$ . The maximum number of such matrices, for which the sum of diagonal elements of $M^{T} M$ is seven, is______.

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Important Questions on Matrices and Determinants

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 24 Feb 2021 Shift 2>Q 49

For the system of linear equations:

$x - 2 y = 1, x - y + k z = - 2, k y + 4 z = 6, k \in R$

Consider the following statements:
(A) The system has unique solution if $k \neq 2, k \neq - 2 .$
(B) The system has unique solution if $k = - 2 .$
(C) The system has unique solution if $k = 2 .$
(D) The system has no-solution if $k = 2 .$
(E) The system has infinite number of solutions if $k \neq - 2 .$

Which of the following statements are correct?

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 24 Feb 2021 Shift 2>Q 50

Let

A

and

B

3 \times 3

real matrices such that

A

is a symmetric matrix and

B

is a skew-symmetric matrix. Then the system of linear equations

(A^{2} B^{2} - B^{2} A^{2}) X = O,

where

X

is a

3 \times 1

column matrix of unknown variables and

O

is a

3 \times 1

null matrix, has

MEDIUM

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IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 25 Feb 2021 Shift 1>Q 51

A = [\begin{matrix} 0 & - \tan (\frac{θ}{2}) \\ \tan (\frac{θ}{2}) & 0 \end{matrix}]

and

(I_{2} + A) {(I_{2} - A)}^{- 1} = [\begin{matrix} a & - b \\ b & a \end{matrix}],

then

13 (a^{2} + b^{2})

is equal to _____ .

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 25 Feb 2021 Shift 1>Q 52

Let

A = [\begin{array}{l} x & y & z \\ y & z & x \\ z & x & y \end{array}],

where

x, y

and

z

are real numbers such that

x + y + z > 0

and

x y z = 2

. If

A^{2} = I_{3}

, then the value of

x^{3} + y^{3} + z^{3}

EASY

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 25 Feb 2021 Shift 1>Q 53

If the system of equations

$k x + y + 2 z = 1$

$3 x - y - 2 z = 2$

$- 2 x - 2 y - 4 z = 3$

has infinitely many solutions, then $k$ is equal to ______ .

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 25 Feb 2021 Shift 2>Q 54

Let

A

be a

3 \times 3

matrix with

det (A) = 4

. Let

R_{i}

denote the

i^{t h}

row of

A

. If a matrix

B

is obtained by performing the operation

R_{2} \to 2 R_{2} + 5 R_{3}

2 A

, then

det (B)

is equal to :

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 25 Feb 2021 Shift 2>Q 55

If for the matrix,

A = [\begin{matrix} 1 & - α \\ α & β \end{matrix}], A A^{T} = I_{2}

, then the value of

α^{4} + β^{4}

is :

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 25 Feb 2021 Shift 2>Q 56

The following system of linear equations

2 x + 3 y + 2 z = 9

3 x + 2 y + 2 z = 9

x - y + 4 z = 8

Let M be any 3×3 matrix with entries from the set 0,1,2. The maximum number of such matrices, for which the sum of diagonal elements of MTM is seven, is______.

Important Questions on Matrices and Determinants

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Let $M$ be any $3 \times 3$ matrix with entries from the set $\{0, 1, 2\}$ . The maximum number of such matrices, for which the sum of diagonal elements of $M^{T} M$ is seven, is______.