1.

Prove that the value of determinant `|{:(1,,omega,,omega^(2)),(omega ,,omega^(2),,1),( omega^(2),, 1,,omega):}|=0` where `omega` is complex cube root of unity .

Answer» `Delta =|{:(1,,omega,,omega^(2)),(omega,,omega^(2),,1),(omega^(2),,1,,omega):}|`
Applying `R_(1) to omega R_(1)` we get
`Delta =(1)/(omega) |{:(omega,,omega^(2),,omega^(3)),(omega,,omega^(2),,1),(omega^(2),,1,,omega):}|`
`=(1)/(omega)|{:(omega,,omega^(2),,omega^(3)),(omega,,omega^(2),,1),(omega^(2),,1,,omega):}|`
`(As R_(1) " and " R_(2) "are identical )"`
`=0`


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