Question

IF $w(\ne 1)$ is a cube root of unity, then
$\left|\begin{array}{ccc}1& 1+i+w^2& w^2\\ 1-i& -1& w^2-1\\ -i& -i+w-1& -1\end{array}\right|$ is equal to

• A. 1
• B. $0$
• C. ω
• D. i

Answer 1

The correct option is A 1

$△=\left|\begin{array}{ccc}1& 1+i+w^2& w^2\\ 1-i& -1& w^2-1\\ -i& -i+w-1& -1\end{array}\right|$

$=\left|\begin{array}{ccc}1& 1-w^2& w^2\\ 1-i& -1& w^2-1\\ -i& -i+w-1& -1\end{array}\right|$ (now $R2\to R2-R1)$

$=0(\because$ Two rows are identical i.e $R2$ and $R3$ becomes identical after the above operation)