Mathematics Crash Course JEE Main>Chapter 19 - Matrices and Determinants>Level 2>Q 1

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Let $B = [\begin{array}{l} 1 & 2 \\ 0 & 1 \end{array}]$ and $A$ be a $2 \times 2$ matrix satisfying ${(A^{T})}^{- 1} = A .$

If $X = A B A^{T},$ then $A^{T} X^{2021} A =$

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Important Questions on Matrices and Determinants

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

If

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

where

i = \sqrt{- 1}

then

[\begin{matrix} z_{1} & z_{2} \\ - \bar{z_{2}} & \bar{z_{1}} \end{matrix}] [\begin{matrix} \bar{z_{1}} & - z_{2} \\ \bar{z_{2}} & z_{1} \end{matrix}] =

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Mathematics Crash Course JEE Main>Chapter 19 - Matrices and Determinants>Level 2>Q 3

Let

a, b

be non zero real numbers such that

a b = 5 / 2

and given

A = [\begin{matrix} a & - b \\ b & a \end{matrix}]

and

A A^{T} = 20 I

(

I

is a unit matrix), then the equation whose roots are

a

and

b

is

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Mathematics Crash Course JEE Main>Chapter 19 - Matrices and Determinants>Level 2>Q 4

If the matrix

A = [\begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 3 & 0 & - 1 \end{matrix}]

satisfies the equation

A^{20} + α A^{19} + β A = [\begin{array}{l} 1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1 \end{array}]

for some real numbers

α

and

β

, then

β - α

is equal to ______.

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Mathematics Crash Course JEE Main>Chapter 19 - Matrices and Determinants>Level 2>Q 5

Let

I

be an identity matrix of order

2 \times 2

and

P = [\begin{matrix} 2 & - 1 \\ 5 & - 3 \end{matrix}]

. Then the value of

n \in N

for which

P^{n} = 5 I - 8 P

is equal to ___ .

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Mathematics Crash Course JEE Main>Chapter 19 - Matrices and Determinants>Level 2>Q 7

Let

A = [\begin{matrix} - 5 & - 8 & 0 \\ 3 & 5 & 0 \\ 1 & 2 & - 1 \end{matrix}]

and

B = A + A^{2} + A^{4} + A^{8} + . . . . + A^{2^{78}}

then sum of all the elements of matrix

B

is equal to

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Mathematics Crash Course JEE Main>Chapter 19 - Matrices and Determinants>Level 2>Q 11

If

A

and

B

are two matrices such that

A B = B

and

B A = A

, then

A^{2} + B^{2}

is equal to

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Mathematics Crash Course JEE Main>Chapter 19 - Matrices and Determinants>Level 2>Q 16

If a square matrix

A

is such that

(A^{T} - \frac{1}{2} I) (A - \frac{1}{2} I) = (A^{T} + \frac{1}{2} I) (A + \frac{1}{2} I) = I,

where

I

is a unit matrix, then

A

is

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

Let

A

be a

n \times n

matrix such that

A

is upper-triangular. Then

adj (A) =