Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 13 - Matrices>Exercises>Q 28

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Let $A, B$ be two matrices different from identify matrix such that $A B = B A$ and $A^{n} - B^{n}$ is invertible for some positive integer $n$ . If $A^{n} - B^{n} = A^{n + 1} - B^{n + 1} = A^{n + 2} - B^{n + 2}$ , then

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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 29

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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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