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3 and 4 .Determinants and Matrices
medium
If $\omega$ is one of the imaginary cube roots of unity, then the value of the determinant $\left| {\begin{array}{*{20}{c}}1&{{\omega ^3}}&{{\omega ^2}}\\ {{\omega ^3}}&1&\omega \\{{\omega ^2}}&\omega &1\end{array}} \right|$ $=$
A
$1$
B
$2$
C
$3$
D
none
Solution
Put $\omega ^3 = 1$ $\left| {\,\begin{array}{*{20}{c}}1&1&{{\omega ^2}}\\ 1&1&\omega \\{{\omega ^2}}&\omega &1\end{array}\,} \right|$ and open by $R_1$ to get $(1 – \omega ^2) + (1 – \omega ) = 3$
Standard 12
Mathematics
