If - and -2 are non-real cube roots of unity- and -1- i- then beginvmatrix 1 - 1- i -lomega- 2 - - 2 1- i - -1 - lomega - 2-1 - i - i - omega -1 -1lendvmatrix -1 A- 1B- - iC- -D- 0

The correct option is D 0R_2 → R_2 R_3 1 amp; 1 + i+ ω^2 amp; ω^2 1 amp; i ω amp; ω^2 i amp; i + ω 1 amp; 1 = 1 amp; i ...

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Byju's Answer

Standard XII

Mathematics

Evaluation of a Determinant

If ω and ω2 a...

Question

If $ω$ and $ω^{2}$ are non-real cube roots of unity, and $\sqrt{- 1} = i$ , then
$∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 + i + ω^{2} & ω^{2} 1 - i & - 1 & ω^{2} - 1 - i & - i + ω - 1 & - 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ =$

1

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- i

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ω

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0

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Solution

The correct option is D $0$
$R_{2} \to R_{2} - R_{3} ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 + i + ω^{2} & ω^{2} 1 & i - ω & ω^{2} - i & - i + ω - 1 & - 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$

$= ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & i - ω & ω^{2} 1 & i - ω & ω^{2} - i & - i + ω - 1 & - 1 \end{matrix} ∣ ∣ ∣ ∣ ∣$

$R_{1}$ and $R_{2}$ are identical.

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Similar questions

Q. If

ω (\neq 1)

is a cube root of unity, then

∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 + i + ω^{2} & ω^{2} 1 - i & - 1 & ω^{2} - 1 - i & - i + ω - 1 & - 1 \end{matrix} ∣ ∣ ∣ ∣ ∣

equals

Q. If

n \neq 3 k

and

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are the cube roots of unity, then

Δ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & ω^{n} & ω^{2 n} ω^{2 n} & 1 & ω^{n} ω^{n} & ω^{2 n} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣

has the value

If $ω \neq 1$ is a cube root of unity and $∣ ∣ ∣ ∣ ∣ \begin{matrix} x + ω^{2} & ω & 1 ω & ω^{2} & 1 + x 1 & x + ω & ω^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0$ , then value of x is

Q. If

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∣ ∣ ∣ ∣ \begin{matrix} 2 & 2 ω & - ω^{2} 1 & 1 & 1 1 & - 1 & 0 \end{matrix} ∣ ∣ ∣ ∣ =

Q. If

i = \sqrt{- 1}

and

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If ω and ω2 are non-real cube roots of unity, and √−1=i, then ∣∣ ∣ ∣∣11+i+ω2ω21−i−1ω2−1−i−i+ω−1−1∣∣ ∣ ∣∣=

The correct option is D 0R2→R2−R3∣∣ ∣ ∣∣11+i+ω2ω21i−ωω2−i−i+ω−1−1∣∣ ∣ ∣∣ =∣∣ ∣ ∣∣1i−ωω21i−ωω2−i−i+ω−1−1∣∣ ∣ ∣∣ R1 and R2 are identical.

If $ω$ and $ω^{2}$ are non-real cube roots of unity, and $\sqrt{- 1} = i$ , then
$∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 + i + ω^{2} & ω^{2} 1 - i & - 1 & ω^{2} - 1 - i & - i + ω - 1 & - 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ =$