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

EASY

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IMPORTANT

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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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Important Questions on Matrices and Determinants

HARD

JEE Main

IMPORTANT

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 ______.

MEDIUM

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IMPORTANT

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 ___ .

MEDIUM

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 19 - Matrices and Determinants>Level 2>Q 6

If

A = [\begin{array}{l} 1 & 0 & 0 \\ x & 1 & 0 \\ x & x & 1 \end{array}]

and

I = [\begin{array}{l} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]

then

A^{3} - 3 A^{2} + 3 A + I

is equal to

HARD

JEE Main

IMPORTANT

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

HARD

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 19 - Matrices and Determinants>Level 2>Q 8

If

P

is a

3 \times 3

matrix such that

P^{T} = 2 P + I,

where

P^{T}

is the transpose of

P

and

I

is the

3 \times 3

identity matrix, then there exists a column matrix

X = [\begin{array}{l} x \\ y \\ z \end{array}] \neq [\begin{array}{l} 0 \\ 0 \\ 0 \end{array}]

such that

HARD

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 19 - Matrices and Determinants>Level 2>Q 9

If

A

is a non singular square matrix where

B = A^{T}

&

A + B^{2} = I

, such that

A^{3} + I = k A

then

k

is equal to

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 19 - Matrices and Determinants>Level 2>Q 10

If

A = [\begin{array}{l} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{array}]

is a root of polynomial

x^{3} - 6 x^{2} + 7 x + k = 0,

then the value of

k

is

MEDIUM

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IMPORTANT

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