Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 13 - Matrices>Exercises>Q 26

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If $A$ and $B$ are symmetric and commute, then which of the following is/are symmetric?

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Important Questions on Matrices

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IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 13 - Matrices>Exercises>Q 27

If

A

and

B

are square matrices of order

3

such that

A^{3} = 8 B^{3} = 8 I

and

\det (A B - A - 2 B + 2 I) \neq 0

, then identify the correct statement (s) , where

I

is identity matrix of order

3

.

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

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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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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IMPORTANT

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