सिद्ध कीजिए की `|{:(,1,omega,omega^(2)),(,omega,omega^(2),1),(,omega^(2),1,omega):}|=0` जहाँ `omega` का मूलधन है

Explanation

बायाँ पक्ष `=|{:(1,omega,omega^(2)),(omega,omega^(2),1),(omega^(2),1,omega):}|`
`=|{:(1+omega+omega^(2),1+omega+omega^(2),1+omega+omega^(2)),(" "omega," "omega^(2)," "1),(" "omega^(2)," "1," "omega):}|`
`R_(1) rarr R_(1) +(R_(2)+R_(3)) "से"," "...(i)`
`because omega=-(1)/(2)+(i sqrt(3))/(2)" "("दिया है")`
`therefore omega^(2) = (-(1)/(2)+(i sqrt(3))/(2))^(2)`
`=(1)/(4)-(3)/(4)=(2i sqrt(3))/(4)`
या `omega^(2) = (1)/(2)-(i sqrt(3))/(2)`
`therefore omega + omega^(2) =-(1)/(2)+(i sqrt(3))/(2)-(1)/(2)-(i sqrt(3))/(2)=-1`
या `omega^(2) + omega + 1 = 0`
`therefore` सारणिक (i) से,
बायाँ पक्ष `=|{:(0,0,0),(omega,omega^(2),1),(omega^(2),1,omega):}|=0" "[because R_(1)"का प्रत्येक अवयव शून्य है"]`
= दायाँ पक्ष |