If `omega!=1` is a complex cube root of unity, then prove that `[{:(1+2omega^(2017)+omega^(2018)," "omega^(2018),1),(1,1+2omega^(2018)+omega^(2017),omega^(2017)),(omega^(2017),omega^(2018),2+2omega^(2017)+omega^(2018)):}]`is singular

Explanation

Let `A=[{:(1+2omega^(2017)+omega^(2018)," "omega^(2018),1),(1,1+2omega^(2018)+omega^(2017),omega^(17)),(omega^(17),omega^(18),2+2omega^(2017)+omega^(2018)):}]`
` therefore" " omega^(3)=1rArr omega^(2017)=omega`
and `omega^(2018)=omega^(2)` then
`[(1+2omega+omega^(2),omega^(2),1),(1,1+omega^(2)+2omega,omega),(omega,omega^(2),2+omega+2omega^(2))]`
`=[(omega,omega^(2),1),(1,omega,omega),(omega,omega^(2),-omega)]" " [therefore1+omega+omega^(2)=0]`
Now, `|A|= [(omega,omega^(2),1),(1,omega,omega),(omega,omega^(2),-omega)]= omega[(omega,omega,1),(1,1,omega),(omega,omega,-omega)]=0 thus, `|A|=0.` Hence, A is singular matrix.