JEE Advance - Mathematics (2011 - Paper 2 Offline - No. 15)
Let $$\omega$$ $$\ne$$ 1 be a cube root of unity and S be the set of all non-singular matrices of the form $$\left[ {\matrix{
1 & a & b \cr
\omega & 1 & c \cr
{{\omega ^2}} & \omega & 1 \cr
} } \right]$$, where each of a, b, and c is either $$\omega$$ or $$\omega$$2. Then the number of distinct matrices in the set S is
2
6
4
8
Explanation
$$\left| A \right| \ne 0$$, as non-singular.
$$\therefore$$ $$\left| {\matrix{ 1 & a & b \cr \omega & 1 & c \cr {{\omega ^2}} & \omega & 1 \cr } } \right| \ne 0$$
$$ \Rightarrow 1(1 - c\omega ) - a(\omega - c{\omega ^2}) + b({\omega ^2} - {\omega ^2}) \ne 0$$
$$ \Rightarrow 1 - c\omega - a\omega + ac{\omega ^2} \ne 0$$
$$ \Rightarrow (1 - c\omega )(1 - a\omega ) \ne 0$$
$$ \Rightarrow a \ne {1 \over \omega },c \ne {1 \over \omega } \Rightarrow a = \omega ,c = \omega $$
and $$b \in \{ \omega ,{\omega ^2}\} \Rightarrow 2$$ solutions
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