Question

Let $w\ne 1$ be a cube roots of unity and S be the set of all non-singular matrices of the form $\left[\begin{array}{ccc}1& a& b\\ \omega & 1& c\\ {\omega }^2& \omega & 1\end{array}\right]$ where each a, b and c is either $\omega$ or ${\omega }^2$. Then the number of district matrices in the set S is?

• A. $2$
• B. $6$
• C. $4$
• D. $8$

Answer 1

Answer: (A)

$2$

a, b, c $\in \{w,w^2\}$
Let A$=\left[\begin{array}{ccc}1& a& b\\ \omega & 1& <\\ {\omega }^2& \omega & 1\end{array}\right]$
$\Rightarrow |A|=1-(a+c)\omega +ac{\omega }^2$
Now $|A|$ will be non-zero only when $a=c=w$
$\therefore (a,b,c)=(\omega ,\omega ,\omega )$ or $(\omega ,{\omega }^2,\omega )$
Hence, number of nm-singular matrices $=2$