Mathematics Crash Course JEE Main>Chapter 19 - Matrices and Determinants>Level 2>Q 17

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IMPORTANT

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Let $A$ be a $n \times n$ matrix such that $A$ is upper-triangular. Then $adj (A) =$

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Important Questions on Matrices and Determinants

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 19 - Matrices and Determinants>Level 2>Q 18

Let

A

be a

3 \times 3

real matrix. If

det (2 Adj (2 Adj (Adj (2 A)))) = 2^{41},

then the value of

det (A^{2})

equals ______.

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Mathematics Crash Course JEE Main>Chapter 19 - Matrices and Determinants>Level 2>Q 19

Let

A

be an invertible matrix. Which of the following is not true?

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 19 - Matrices and Determinants>Level 2>Q 20

$|\begin{array}{l} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{array}| + |\begin{array}{l} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{array}| + \dots \dots + |\begin{matrix} n & 0 & 0 \\ 0 & n + 1 & 0 \\ 0 & 0 & n + 2 \end{matrix}|$ is equal to

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 19 - Matrices and Determinants>Level 2>Q 21

If

Δ = |\begin{matrix} 1 & a & a^{2} \\ \cos (n - 1) x & \cos n x & \cos (n + 1) x \\ \sin (n - 1) x & \sin n x & \sin (n + 1) x \end{matrix}|

, then

Δ

is

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 19 - Matrices and Determinants>Level 2>Q 22

If

A, B

and

C

are the angles of a triangle, then the value of

Δ = |\begin{matrix} - 1 & \cos C & \cos B \\ \cos C & - 1 & \cos A \\ \cos B & \cos A & - 1 \end{matrix}|

is

HARD

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 19 - Matrices and Determinants>Level 2>Q 23

ω \neq 1

is cube root of unity and

x + y + z \neq 0

, then

|\begin{matrix} \frac{x}{1 + ω} & \frac{y}{ω + ω^{2}} & \frac{z}{ω^{2} + 1} \\ \frac{y}{ω + ω^{2}} & \frac{z}{ω^{2} + 1} & \frac{x}{1 + ω} \\ \frac{z}{ω^{2} + 1} & \frac{x}{1 + ω} & \frac{y}{ω + ω^{2}} \end{matrix}| = 0

, if

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Mathematics Crash Course JEE Main>Chapter 19 - Matrices and Determinants>Level 2>Q 24

Let

P = {[p_{i j}]}_{3 \times 3}

and

Q = {[q_{i j}]}_{3 \times 3}

be such that

d e t (P) = 4

and

q_{i j} = 2^{i + j} p_{i j} (1 \leq i, j \leq 3)

, then the value of

\frac{5}{2^{15}} d e t (Q)

is

HARD

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 19 - Matrices and Determinants>Level 2>Q 25

The value of determinant

|\begin{matrix} 2 a_{1} b_{1} & a_{1} b_{2} + a_{2} b_{1} & a_{1} b_{3} + a_{3} b_{1} \\ a_{1} b_{2} + a_{2} b_{1} & 2 a_{2} b_{2} & a_{2} b_{3} + a_{3} b_{2} \\ a_{1} b_{3} + b_{1} a_{3} & a_{2} b_{3} + a_{3} b_{2} & 2 a_{3} b_{3} \end{matrix}|

is (where

a_{i}, b_{i} > 0

for

i = 1, 2, 3

)