If ${\rm{\alpha }},{\rm{\beta }},{\rm{\gamma }}$ are the cube roots of unity, then the value of the determinant $\left| {\begin{array}{*{20}{c}} {{e^\alpha }}&{{e^{2\alpha }}}&{\left( {{e^{3\alpha }} - 1} \right)}\\ {{e^\beta }}&{{e^{2\beta }}}&{\left( {{e^{3\beta }} - 1} \right)}\\ {{e^\gamma }}&{{e^{2\gamma }}}&{\left( {{e^{3\gamma }} - 1} \right)} \end{array}} \right|$ is equal to

JEE MATHEMATICS MCQ: MATRICES

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\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma }

are the cube roots of unity, then the value of the determinant

\left| {\begin{array}{cccccccccccccccccccc} {{e^\alpha }}&{{e^{2\alpha }}}&{\left( {{e^{3\alpha }} - 1} \right)}\\ {{e^\beta }}&{{e^{2\beta }}}&{\left( {{e^{3\beta }} - 1} \right)}\\ {{e^\gamma }}&{{e^{2\gamma }}}&{\left( {{e^{3\gamma }} - 1} \right)} \end{array}} \right|

is equal to

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Correct option: C

Option C (0) is correct because by simplifying the determinant using properties of cube roots of unity, the determinant evaluates to 0.

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\left|\begin{array}{ccc}a & b-c & b+c \\ c+a & b & c-a \\ a-b & a+b & c\end{array}\right|=0

a, b, c

a x+b y+c=0

{x_2} - {x_3} = 1,\; - {x_1} + 2{x_3} = - 2,\;{x_1} - 2,\;{x_1} - 2{x_2} = 3

2x + 3y + 1 = 0,3x + y - 2 = 0

ax + 2y - b = 0

\left|\begin{array}{ccc}1 & \left(\alpha^{2 x}-\alpha^{-2 x}\right)^2 & \left(\alpha^{2 x}+\alpha^{-2 x}\right)^2 \\ 1 & \left(\beta^{2 x}-\beta^{-2 x}\right)^2 & \left(\beta^{2 x}+\beta^{-2 x}\right)^2 \\ 1 & \left(\gamma^{2 x}-\gamma^{-2 x}\right)^2 & \left(\gamma^{2 x}+\gamma^{-2 x}\right)^2\end{array}\right|

\omega

\left| {\begin{array}{cccccccccccccccccccc} 1&\omega &{{\omega ^2}}\\ \omega &{{\omega ^2}}&1\\ {{\omega ^2}}&1&\omega \end{array}} \right|

A

2 \times 2

A(\operatorname{adj} A)=\left[\begin{array}{cc}10 & 0 \\ 0 & 10\end{array}\right]

|A|