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Let $\omega\neq 1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form $\begin{vmatrix}1&a&b\\\omega&1&c\\\omega^2&\omega&1\end{vmatrix}$ where each of a,b and c is either $\omega$ or $\omega^2$.Then the number of distinct matrices in the set S is


1 Answer

For the given matrix to be non-singular $\begin{vmatrix}1 &a&b\\\omega&1&c\\\omega^2&\omega&1\end{vmatrix}\neq 0$
$\Rightarrow 1-(a+c)\omega+ac\omega^2\neq 0$
$\Rightarrow (1-a\omega)(1-c\omega)\neq 0$
$\Rightarrow a\neq \omega^2$ and $c\neq \omega^2$
Where $\omega$ is complex cube root of unity.
As $a,b$ and $c$ are complex cube root of unity.
$\therefore a$ and $c$ can take 2 values (i.e) $\omega$ and $\omega^2$
$\therefore$ Total number of distinct matrices=$1\times 1\times 2$
$\Rightarrow 2$
Hence (a) is the correct answer.
answered Nov 20, 2013 by sreemathi.v

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