Question
Let $$\omega \ne 1$$ be a cube root of unity and $$S$$ be the set of all non-singular matrices of the form \[\left| \begin{array}{l}
1\,\,\,\,\,\,\,\,a\,\,\,\,\,\,\,\,b\\
\omega \,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,c\\
{\omega ^2}\,\,\,\,\omega \,\,\,\,\,\,\,\,1
\end{array} \right|\] where each of $$a, b$$ and $$c$$ is either \[\omega \] or \[{\omega ^2}\] . Then the number of distinct matrices in the set $$S$$ is
A.
2
B.
6
C.
4
D.
8
Answer :
6
Solution :
For the given matrix to be non - singular
\[\left| \begin{array}{l}
1\,\,\,\,\,\,\,\,a\,\,\,\,\,\,\,\,b\\
\omega \,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,c\\
{\omega ^2}\,\,\,\,\omega \,\,\,\,\,\,\,\,1
\end{array} \right| \ne 0\]
$$\eqalign{
& \Rightarrow \,\,1 - \left( {a + c} \right)\omega + a\,c\,{\omega ^2} \ne 0 \cr
& \Rightarrow \,\,\left( {1 - a\,\omega } \right)\left( {1 - c\,\omega } \right) \ne 0 \cr} $$
$$ \Rightarrow \,\,a \ne {\omega ^2}\,\,{\text{and }}c \ne {\omega ^2}$$ where $$\omega $$ is complex cube root of unity.
As $$a, b$$ and $$c$$ are complex cube roots of unity
∴ $$a$$ and $$c$$ can take only one value i.e. $$\omega $$ while $$b$$ can take 2 values i.e. $$\omega $$ and $${\omega ^2}.$$
∴ Total number of distinct matrices $$ = 1 \times 1 \times 2 = 2$$