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Multiplication of Matrices

Let n ≥ 2 b...

Question

Let $n \geq 2$ be an integer.
$A = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} cos (\frac{2 π}{n}) & sin (\frac{2 π}{n}) & 0 - sin (\frac{2 π}{n}) & sin (\frac{2 π}{n}) & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$ and $I$ is the identify matrix of order $3$ ., then following of which is correct

A

A^{n} + I

and

A^{n - 1} \neq I

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B

A^{m} \neq I

for any positive integer

m

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C

A

is not invertible

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D

A

is invertible

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Solution

The correct option is A $A^{n} + I$ and $A^{n - 1} \neq I$

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Similar questions

n \geq 2

A = ⎛ ⎜ ⎝ \begin{matrix} cos (2 π / n) & sin (2 π / n) & 0 sin (2 π / n) & cos (2 π / n) & 0 0 & 0 & 1 \end{matrix} ⎞ ⎟ ⎠

I

is the identity matrix of order

3

n \geq 2

A = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} cos (\frac{2 π}{n}) & sin (\frac{2 π}{n}) & 0 - sin (\frac{2 π}{n}) & cos (\frac{2 π}{n}) & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

I

is the identity matrix of order

3

., then following of which is correct

A

B

be two matrices different from

I

A B = B A

A^{n} - B^{n}

is invertible for some positive integer

n

A^{n} - B^{n} = A^{n + 1} - B^{n + 1} = A^{n + 2} - B^{n + 2}

3 \times 3

M

| M |

denote the determinant of

M

I

3 \times 3

identify matrix. Let

E

F

3 \times 3

matrices such that

(I - E F)

is invertible. If

G = (I - E F)^{- 1}

, then which of the following statements is(are) TRUE?

P

is a non null matrix of order

3

with real number as entries, such that

P^{3} = 0,

O

is null matrix of order

3

I

is the identity matrix of order

3

)

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