Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 13 - Matrices>Exercises>Q 29

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Let $A$ and $B$ be square matrices of the same order such that $A^{2} = I$ and $B^{2} = I$ , then which of the following is CORRECT?

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Important Questions on Matrices

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 13 - Matrices>Exercises>Q 30

Let

B

is an invertible square matrix and

B

is the adjoint of matrix

A

such that

A B = B^{T}

. Then

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 13 - Matrices>Exercises>Q 31

First row of a matrix

A

is

[1 3 2]

. If

adj A = [\begin{matrix} - 2 & 4 & α \\ - 1 & 2 & 1 \\ 3 α & - 5 & - 2 \end{matrix}]

, then a

\det (A)

is

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 13 - Matrices>Exercises>Q 32

Let

A

be a square matrix of order

3

satisfies the relation

A^{3} - 6 A^{2} + 7 A - 8 I = O

and

B = A - 2 I

. Also,

\det (A) = 8

, then

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 13 - Matrices>Exercises>Q 33

Which of the following matrices have eigen values as

1

and

- 1

?

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 13 - Matrices>Exercises>Q 1

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

and

(A^{8} + A^{6} + A^{4} + A^{2} + I) V = [\begin{matrix} 0 \\ 11 \end{matrix}]

(where

I

is the

2 \times 2

identity matrix), then the product of all elements of matrix

V

is equal to

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 13 - Matrices>Exercises>Q 2

If

[\begin{matrix} a & b \\ c & 1 - a \end{matrix}]

is an idempotent matrix and

f (x) = x - x^{2} & b c = \frac{1}{4}

, then the value of

f (a)

is,

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 13 - Matrices>Exercises>Q 3

Let

X

be the solution set of the equation

A^{x} = I,

where

A = [\begin{matrix} 0 & 1 & - 1 \\ 4 & - 3 & 4 \\ 3 & - 3 & 4 \end{matrix}]

and

I

is the corresponding unit matrix and

x \in N,

then the minimum value of

\underset{}{\sum_{x}} (\cos^{x} θ + \sin^{x} θ), θ \in (0, \frac{π}{2}),

is

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 13 - Matrices>Exercises>Q 4

A = [\begin{matrix} 1 & \tan x \\ - \tan x & 1 \end{matrix}], f (x)

is defined as

f (x) = \det (A^{T} A^{- 1})

, then the value of

\underset{n times}{\underset{⏟}{f (f (f (f . . . . . . . . . . f (x))))}} (n \geq 2)

is: