Let `omega` be a complex number such that `2omega+1=z` where `z=sqrt(-3)`. If `|{:(1,1,1),(1,-omega^(2)-1,omega^(2)),(1,omega^(2),omega^(7))|=3k`, then k is equal to

Let `omega` be a complex number such that `2omega+1=z` where `z=sqrt(-

1.	Let `omega` be a complex number such that `2omega+1=z` where `z=sqrt(-3)`. If `\|{:(1,1,1),(1,-omega^(2)-1,omega^(2)),(1,omega^(2),omega^(7))\|=3k`, then k is equal to
Answer» Here, `z = sqrt (-3) = sqrt3i` `:. 2omega+1 = sqrt3i` `=>omega = (sqrt3i-1)/2` Now, we will find the value of the given determinant. `\|[1,1,1],[1,-omega^2-1,omega^2],[1,omega^2,omega^7]\|` Now, we know, `1+omega+omega^2 = 0` amd `omega^3 = 1`So, our determinant becomes, `\|[1,1,1],[1,omega,omega^2],[1,omega^2,omega]\|` `=[omega^2-omega^4-omega+omega^2+omega^2-omega]` `=[omega^2-omega-omega+omega^2+omega^2-omega]` `=3(omega^2 - omega)` `=3(-1/2-(sqrt3i)/2 + 1/2 - (sqrt3i)/2))` `= -3sqrt3i` `:. -3sqrt3i = 3k` `=> k = -sqrt3i` `=> k = -z.`