Alpha Question Bank for Engineering: Mathematics>Chapter 9 - Matrices and Determinants>Exercise-2>Q 52

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Let $A, B, C, D$ be real matrices such that $A^{⊤} = B C D; B^{⊤} = C D A; C^{⊤} = D A B$ and $D^{⊤} = A B C$ for the matrix $M = A B C D,$ then find $M^{2016} ?$

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Important Questions on Matrices and Determinants

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 9 - Matrices and Determinants>Exercise-2>Q 53

Let

A

and

B

be two

2 \times 2

matrix with real entries. If

A B = O

and

tr (A) = tr (B) = 0

then

HARD

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Alpha Question Bank for Engineering: Mathematics>Chapter 9 - Matrices and Determinants>Exercise-2>Q 54

If

A^{- 1} = [\begin{matrix} 1 & - 1 & 0 \\ 0 & - 2 & 1 \\ 0 & 0 & - 1 \end{matrix}],

then

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Alpha Question Bank for Engineering: Mathematics>Chapter 9 - Matrices and Determinants>Exercise-2>Q 55

If

A

and

B

are the square matrices of order

3,

then the true statement is

/

are (where

I

is unit matrix).

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Alpha Question Bank for Engineering: Mathematics>Chapter 9 - Matrices and Determinants>Exercise-2>Q 56

Let

M

be a

3 \times 3

non-singular matrix with

det (M) = 4 .

If

M^{- 1} adj (adj M) = k^{2} I,

then the value of

' k'

may be:

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Alpha Question Bank for Engineering: Mathematics>Chapter 9 - Matrices and Determinants>Exercise-2>Q 57

If

A X = B,

where

A

is

3 \times 3

and

X

and

B

are

3 \times 1

matrices, then which of the following is correct?

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Alpha Question Bank for Engineering: Mathematics>Chapter 9 - Matrices and Determinants>Exercise-3>Q 2

Let

ω

be a complex number such that

2 ω + 1 = z,

where

z = \sqrt{- 3}

. If |\begin{matrix} 1 & 1 & 1 \\ 1 & - ω^{2} - 1 & ω^{2} \\ 1 & ω^{2} & ω^{7} \end{matrix}| = 3 k

, then

k

is equal to

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Alpha Question Bank for Engineering: Mathematics>Chapter 9 - Matrices and Determinants>Exercise-3>Q 7

Let

M

and

N

be two

2 n \times 2 n

non-singular skew-symmetric matrices such that

M N = N M .

If

P^{T}

denotes the transpose of

P,

then

M^{2} N^{2} {(M^{T} N)}^{- 1} {(M N^{- 1})}^{T}

is equal to

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Alpha Question Bank for Engineering: Mathematics>Chapter 9 - Matrices and Determinants>Exercise-3>Q 9

Let

ω \neq 1

be a cube root of unity and

S

be the set of all non-singular matrices of the form

[\begin{array}{l} 1 & a & b \\ ω & 1 & c \\ ω^{2} & ω & 1 \end{array}]

, where each of

a, b

and

c

is either

ω

or

ω^{2} .

Then the number of distinct matrices in the set

S

is