# $If w\left ( \ne 1 \right )\, \, \, is\, \, \, a\, \, \, cube\, \, \, root \, \, \,of \, \, \,unity\, \, \, then$$\begin{vmatrix} 1 & 1+i+w^{2} & w^{2} \\ 1-i &-1 &w^{2}-1\\ -i& -i+w-1 & -1\end{vmatrix} =$ Option 1) 0 Option 2) 1 Option 3) i Option 4) w

As leant in concept

Value of determinants of order 3 -

-

Sum of the cube roots of unity is zero.

$1+ \omega +\omega^{2}= 0$

$\begin{vmatrix} 1 &i-\omega &\omega^{2} \\ 1-i&-1 &\omega^{2}-1 \\ -i&-i+\omega+1 &-1 \end{vmatrix}$

$1(1+i\omega^{2}-1+ \omega^{^{2}}-i +\omega-1) - (i-\omega)(-1+i+i\omega^{2}-1) +\omega^{2}[(1-i)(-i+\omega-1)-i]$

On simplifying, the result will be zero

Option 1)

0

Correct option

Option 2)

1

Incorrect option

Option 3)

i

Incorrect option

Option 4)

w

Incorrect option

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