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

MEDIUM

JEE Main

IMPORTANT

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The number of elements in the set { $A = [\begin{matrix} a & b \\ 0 & d \end{matrix}] : a, b, d \in {- 1, 0, 1}$ and $(I - A)^{3} = I - A^{3}$ }, where $I$ is $2 \times 2$ identity matrix, is .

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Important Questions on Matrices and Determinants

EASY

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 1 September 2021 Shift 2>Q 14

Consider the system of linear equations

$- x + y + 2 z = 0$

$3 x - a y + 5 z = 1$

$2 x - 2 y - a z = 7$

Let $S_{1}$ be the set of all $a \in R$ for which the system is inconsistent and $S_{2}$ be the set of all $a \in R$ for which the system has infinitely many solutions. If $n (S_{1})$ and $n (S_{2})$ denote the number of elements in $S_{1}$ and $S_{2}$ respectively, then

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 20 July 2021 Shift 1>Q 15

Let

A = [\begin{array}{l} 2 & 3 \\ a & 0 \end{array}], a \in R

be written as

P + Q

where

P

is a symmetric matrix and

Q

is skew symmetric matrix. If

det (Q) = 9

, then the modulus of the sum of all possible values of determinant of

P

is equal to:

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 20 July 2021 Shift 1>Q 16

Let

A = [\begin{matrix} \begin{matrix} 1 & - 1 & 0 \\ 0 & 1 & - 1 \\ 0 & 0 & 1 \end{matrix} \end{matrix}]

and

B = 7 A^{20} - 20 A^{7} + 2 I

,
where

I

is an identity matrix of order

3 \times 3 .

B = [b_{i j}]

, then

b_{13}

is equal to

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 20 July 2021 Shift 1>Q 17

Let

a, b, c, d

be in arithmetic progression with common difference

λ

. If

|\begin{array}{l} x + a - c & x + b & x + a \\ x - 1 & x + c & x + b \\ x - b + d & x + d & x + c \end{array}| = 2

, then value of

λ^{2}

is equal to________.

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 20 July 2021 Shift 2>Q 18

The value of $k \in R,$ for which the following system of linear equations

$3 x - y + 4 z = 3$

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

$6 x + 5 y + k z = - 3$

has infinitely many solutions, is:

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 20 July 2021 Shift 2>Q 19

Let

A = \{a_{i j}\}

be a

3 \times 3

matrix, where

a_{i j} = \{\begin{matrix} {(- 1)}^{j - i} if i < j \\ 2 if i = j \\ {(- 1)}^{i + j} if i > j \end{matrix}

then

\det (3 Adj (2 A^{- 1}))

is equal to ________.

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 22 July 2021 Shift 1>Q 20

The values of

λ

and

μ

such that the system of equations

x + y + z = 6, 3 x + 5 y + 5 z = 26

and

x + 2 y + λ z = μ

has no solution, are:

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 22 July 2021 Shift 1>Q 21

Let

A = [a_{i j}]

be a real matrix of order

3 \times 3,

such that

a_{i 1} + a_{i 2} + a_{i 3} = 1,

for

i = 1, 2, 3 .

Then, the sum of all the entries of the matrix

A^{3}

is equal to:

The number of elements in the set {A=ab0d: a, b, d∈{-1, 0, 1} and (I-A)3=I-A3}, where I is 2×2 identity matrix, is .

Important Questions on Matrices and Determinants

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The number of elements in the set { $A = [\begin{matrix} a & b \\ 0 & d \end{matrix}] : a, b, d \in {- 1, 0, 1}$ and $(I - A)^{3} = I - A^{3}$ }, where $I$ is $2 \times 2$ identity matrix, is .