Question
If $$1,\omega ,{\omega ^2}$$ are the cube roots of unity, then \[\Delta = \left| \begin{array}{l}
\,1\,\,\,\,\,\,\,\,\,\,\,\,{\omega ^n }\,\,\,\,\,\,\,\,\,\,{\omega ^{2n}}\\
{\omega ^n}\,\,\,\,\,\,\,\,\,{\omega ^{2n}}\,\,\,\,\,\,\,\,\,1\\
{\omega ^{2n}}\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,{\omega ^n}
\end{array} \right|\] is equal to
A.
\[{\omega ^2}\]
B.
0
C.
1
D.
\[{\omega}\]
Answer :
0
Solution :
\[\Delta = \left| \begin{array}{l}
\,1\,\,\,\,\,\,\,\,\,\,\,\,{\omega ^n }\,\,\,\,\,\,\,\,\,\,{\omega ^{2n}}\\
{\omega ^n}\,\,\,\,\,\,\,\,\,{\omega ^{2n}}\,\,\,\,\,\,\,\,\,1\\
{\omega ^{2n}}\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,{\omega ^n}
\end{array} \right|\]
$$\eqalign{
& = 1\left( {{\omega ^{3n}} - 1} \right) - {\omega ^n}\left( {{\omega ^{2n}} - {\omega ^{2n}}} \right) + {\omega ^{2n}}\left( {{\omega ^n} - {\omega ^{4n}}} \right) \cr
& = {\omega ^{3n}} - 1 - 0 + {\omega ^{3n}} - {\omega ^{6n}} \cr
& = 1 - 1 + 1 - 1 = 0\,\left[ {\because \,\,{\omega ^{3n}} = 1} \right] \cr} $$