If `|{:(x,y,0),(0,x,y),(y,0,x):}|=0`, then which one of the following is correct ?

Options

A. `(x)/(y)` is one of the cube roots of unity
B. x is one of the cube roots of unity
C. y is one of the cube roots of unity
D. `(x)/(y)` is one of the cube roots of -1

Answer

D

Explanation

`|{:(x,y,0),(0,x,y),(y,0,x):}|=0 rArr x(x^(2)-0)-y(0-y^(2))+0 = 0`
`rArr x^(3)+y^(3)=0`
`rArr x^(3)=-y^(3)`
`rArr (x^(3))/(y^(3))=-1 rArr ((x)/(y))^(3)=-1 rArr (x)/(y)=sqrt(-1)`