Let `omega` be the complex number `cos((2pi)/3)+isin((2pi)/3)`. Then the number of distinct complex cos numbers z satisfying `Delta=|(z+1,omega,omega^2),(omega,z+omega^2,1),(omega^2,1,z+omega)|=0` is

Answer

1

Explanation

`omega =e^(i2pi//3)`
`|{:(z+1,,omega,,omega^(2)),(omega,,z+omega^(2),,1),(omega^(2),,1,,z+omega):}|=0`
`z |{:(1,,omega,,omega^(2)),(1,,z+omega^(2),,1),(1,,1,,z+omega):}|=0`
`"(Applying " C_(1) to C_(1) +C_(2) +C_(3) " and using " 1+omega +omega^(2) =0")"`
`" or " z[(z+omega^(2)) (z+omega)-1-omega(z+omega-1) +omega^(2)(1-z-omega^(2))] =0`
`" or " z^(3) =0`
`" or " z=0` is only solutions.