(A) Let $\Delta = \left| \begin{array}{ccc} 1 & \omega & -\omega^2/2 \\ 1 & 1 & 1 \\ 1 & -1 & 0 \end{array} \right|$.Expanding along the first column:

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(A) Let $\Delta = \left| \begin{array}{ccc} 1 & \omega & -\omega^2/2 \\ 1 & 1 & 1 \\ 1 & -1 & 0 \end{array} \right|$.Expanding along the first column:$\Delta = 1(0 - (-1)) - 1(0 - \omega^2/2) + 1(\omega - (- \omega^2/2))$$= 1(1) - 1(-\omega^2/2) + 1(\omega + \omega^2/2)$$= 1 + \omega^2/2 + \omega + \omega^2/2$$= 1 + \omega + \omega^2$.Since $\omega$ is a complex cube root of unity,we know that $1 + \omega + \omega^2 = 0$.Therefore,$\Delta = 0$.

Class 12 Mathematics 3 and 4 .Determinants and Matrices Expansion of determinants, Solution of equation in the form of determinants and area of triangle and Equation of Line

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If $\omega$ is a complex cube root of unity,then $\left| \begin{array}{ccc} 1 & \omega & -\omega^2/2 \\ 1 & 1 & 1 \\ 1 & -1 & 0 \end{array} \right| = $

A
$0$
B
$1$
C
$\omega$
D
$\omega^2$

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3 and 4 .Determinants and Matrices

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Expansion of determinants, Solution of equation in the form of determinants and area of triangle and Equation of Line

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If $\omega$ is a complex cube root of unity,then $\left| \begin{array}{ccc} 1 & \omega & -\omega^2/2 \\ 1 & 1 & 1 \\ 1 & -1 & 0 \end{array} \right| = $

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