EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 25 June 2022 Shift 1>Q 159

HARD

JEE Main

IMPORTANT

Earn 100

Let $A$ be a $3 \times 3$ real matrix such that $A (\begin{array}{l} 1 \\ 1 \\ 0 \end{array}) = (\begin{array}{l} 1 \\ 1 \\ 0 \end{array}); A (\begin{array}{l} 1 \\ 0 \\ 1 \end{array}) = (\begin{matrix} - 1 \\ 0 \\ 1 \end{matrix})$ and $A (\begin{array}{l} 0 \\ 0 \\ 1 \end{array}) = (\begin{array}{l} 1 \\ 1 \\ 2 \end{array})$ . If $X = {[x_{1} x_{2} x_{3}]}^{T}$ and $I$ is an identity matrix of order $3$ , then the system $(A - 2 I) X = (\begin{array}{l} 4 \\ 1 \\ 1 \end{array})$ has

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Important Questions on Matrices and Determinants

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 25 June 2022 Shift 2>Q 162

Let

A = (\begin{array}{l} 2 & - 2 \\ 1 & - 1 \end{array})

and

B = (\begin{matrix} - 1 & 2 \\ - 1 & 2 \end{matrix})

. Then the number of elements in the set

{(n, m) : n, m \in \{1, 2, \dots \dots . 10\}

and

n A^{n} + m B^{m} = I}

is _____.

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 26 June 2022 Shift 2>Q 164

Let $X = [\begin{array}{l} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}], Y = α I + β X + γ X^{2}$ and $Z = α^{2} I - α β X + (β^{2} - α γ) X^{2}, α, β, γ \in ℝ$ .

If $Y^{- 1} = [\begin{matrix} \frac{1}{5} & \frac{- 2}{5} & \frac{1}{5} \\ 0 & \frac{1}{5} & \frac{- 2}{5} \\ 0 & 0 & \frac{1}{5} \end{matrix}]$ , then ${(α - β + γ)}^{2}$ is equal to ______.

EASY

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 26 June 2022 Shift 1>Q 165

Let

A

be a

3 \times 3

invertible matrix. If

|adj (24 A)| =

adj (3 adj (2 A)) |

, then

{|A|}^{2}

is equal to

EASY

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 27 June 2022 Shift 1>Q 168

The positive value of the determinant of the matrix

A

, whose

A d j (A d j (A)) = [\begin{matrix} 14 & 28 & - 14 \\ - 14 & 14 & 28 \\ 28 & - 14 & 14 \end{matrix}]

, is ______.

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 27 June 2022 Shift 2>Q 170

Let

A

and

B

be two

3 \times 3

matrices such that

A B = I

and

|A| = \frac{1}{8}

then

|a d j (B a d j (2 A))|

is equal to

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 28 June 2022 Shift 1>Q 172

Let

A

be a matrix of order

3 \times 3

and

\det (A) = 2

. Then

\det (det (A) adj (5 adj (A^{3}))

is equal to _____.

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 28 June 2022 Shift 2>Q 173

Let

A = (\begin{matrix} 1 + i & 1 \\ - i & 0 \end{matrix})

where

i = \sqrt{- 1}

. Then, the number of elements in the set

\{n \in {1, 2, \dots ., 100} : A^{n} = A\}

is

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Matrices and Determinants>JEE Main - 29 June 2022 Shift 1>Q 175

Let

A = [a_{i j}]

be a square matrix of order

3

such that

a_{i j} = 2^{j - i}

, for all

i, j = 1, 2, 3

. Then, the matrix

A^{2} + A^{3} + \dots + A^{10}

is equal to