Let - be a root of the equation x 2- x -1-0 and the matrix A -1-3 beginbmatrix1 - 1 - 1 1 - - alpha - 2 1lend bmatrix - then the matrix A 31 is equal toA- A2B- A 3C- AD- I 3

The correct option is B A^3The roots of equation x^2 + x + 1 = 0 are complex cube roots of unity. ∴α = ω or ω^2 A = 1√(3)[ 1 amp; 1 amp; 1; nbsp; ...

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Byju's Answer

Standard XII

Mathematics

Multiplication of Matrices

Let α be a ro...

Question

Let $α$ be a root of the equation $x^{2} + x + 1 = 0$ and the matrix
$A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 1 & α & α^{2} 1 & α^{2} & α^{4} \end{matrix} ⎤ ⎥ ⎦$ then the matrix $A^{31}$ is equal to

A

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A^{2}

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A^{3}

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I_{3}

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Solution

The correct option is C $A^{3}$

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Similar questions

Q. Let

α

be a root of the equation

x^{2} + x + 1 = 0

and the matrix

A = \frac{1}{\sqrt{3}}

⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 1 & α & α^{2} 1 & α^{2} & α^{4} \end{matrix} ⎤ ⎥ ⎦

, then the matrix

A^{31}

is equal to

Q. If

α

and

β

are the roots of the equation

1 + x + x^{2} = 0

, then the matrix product

[\begin{matrix} 1 & β α & α \end{matrix}] [\begin{matrix} α & β 1 & β \end{matrix}]

is equal to

$A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 2 0 & 2 & 1 1 & 0 & 2 \end{matrix} ⎤ ⎥ ⎦ a n d A^{3} = (a A - I) (b A - I)$ , where a, b are integers and I is a $3 \times 3$ unit matrix then value of (a + b) is equal to

Q. Let

A = ⎡ ⎢ ⎣ \begin{matrix} 2 & 0 & 7 0 & 1 & 0 1 & - 2 & 1 \end{matrix} ⎤ ⎥ ⎦

and

⎡ ⎢ ⎣ \begin{matrix} - α & 14 α & 7 α 0 & 1 & 0 α & - 4 α & - 2 α \end{matrix} ⎤ ⎥ ⎦

. If

A B = I

, where

I

is an identity matrix of order

3

then trace

B

has value equal to

Q. If the matrix

A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 2 & 0 3 & 0 & - 1 \end{matrix} ⎤ ⎥ ⎦

satisfies the equation

A^{20} + α A^{19} + β A = A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 4 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦

for some real numbers

α

and

β

, then

β - α

is equal to

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Let α be a root of the equation x2+x+1=0 and the matrixA=⎡⎢⎣1111αα21α2α4⎤⎥⎦ then the matrix A31 is equal to

The correct option is C A3

Let $α$ be a root of the equation $x^{2} + x + 1 = 0$ and the matrix
$A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 1 & α & α^{2} 1 & α^{2} & α^{4} \end{matrix} ⎤ ⎥ ⎦$ then the matrix $A^{31}$ is equal to

The correct option is C $A^{3}$