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If `omega`is an imaginary cube root of unity, then the value of the determinant `|(1+omega,omega^2,-omega),(1+omega^2,omega,-omega^2),(omega+omega^2,omega,-omega^2)|`

Text Solution

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`omega^2 + omega + 1 = 0`
so, changing the determinant
`|(-omega^2, omega , - omega),(-omega, omega ,-omega^2),(-1 , omega, -omega^2)|`
`= (omega^5 + omega^3+ omega^4) - (omega^2 + omega^5 + omega^5)`
`= (omega^2+1+ omega) - (omega^2 + omega^2 + omega^2)`
`= 0 - (3omega^2)`
`= - 3omega^2`
answer
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