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

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IMPORTANT

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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

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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 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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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

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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

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Mathematics Crash Course JEE Main>Chapter 19 - Matrices and Determinants>Level 2>Q 12

If

A = [\begin{array}{l} 1 & 4 & 4 \\ 4 & 1 & 4 \\ 4 & 4 & 1 \end{array}]

and

A^{- 1}

exists and not equal to zero, then

(A^{2} - 8 A) A^{- 1} =

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Mathematics Crash Course JEE Main>Chapter 19 - Matrices and Determinants>Level 2>Q 13

If

B, C

are square matrices of order

n

and if

A = B + C, B C = C B, C^{2} = O,

then for any positive integer

p, A^{p + 1} = B^{k} [B + (p + 1) C]

, where

k =

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Mathematics Crash Course JEE Main>Chapter 19 - Matrices and Determinants>Level 2>Q 14

A square matrix

A = [\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}],

then which of the following statement is not correct?

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 19 - Matrices and Determinants>Level 2>Q 15

If matrix

A = {[a_{i j}]}_{3 \times 3},

matrix

B = {[b_{i j}]}_{3 \times 3},

where

a_{i j} + a_{j i} = 0

and

b_{i j} - b_{j i} = 0 \forall i, j

, then

A^{4} \cdot B^{3}

is

EASY

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IMPORTANT

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