Question

Let $\omega$ be a complex number such that $2 \omega + 1 = z$ , where $z = \sqrt{-3}. If \ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ [tex] \left | \begin{array}{ccc} 1 & 1 & 1 \\ 1 & - {\omega}^{2} -1 & {\omega}^{2} \\ 1 & {\omega}^{2} & {\omega}^{7} \end{array} \right |$ = 3k, then k is equal to​

Answer 1

$\underline{ \underline{ \sf \red{Question}}} \red{: }$

Let $\omega$ be a complex number such that $2 \omega + 1 = z$ , where $z = \sqrt{-3}$. If $\left | \begin{array}{ccc} 1 & 1 & 1 \\ 1 & - {\omega}^{2} -1 & {\omega}^{2} \\ 1 & {\omega}^{2} & {\omega}^{7} \end{array} \right |$ = 3k, then k is equal to

$\underline{ \underline{ \sf \red{Answer}}} \red{: }$

$\rm \: Given$

$\rm \: 2 \omega + 1 = z$

$\rm \implies \: 2 \omega + 1 = \sqrt{ - 3} \: \: \: \: \: \: \: [ \because \: z \: = \: \sqrt{ - 3} ]$

$\rm \implies \omega = \dfrac{ - 1 + \sqrt{3} \: i}{2}$

$\rm \: Since, \: \omega \: is \: cube \: root \: of \: unity$

$\rm \therefore \: { \omega}^{2} = \dfrac{ - 1 - \sqrt{3} \: i }{2} \: \: \: and \: \: { \omega}^{3n} = 1$

$\rm \: Now, \left | \begin{array}{ccc} 1 & 1 & 1 \\ 1 & - {\omega}^{2} -1 & {\omega}^{2} \\ 1 & {\omega}^{2} & {\omega}^{7} \end{array} \right | \: = 3k$

$\rm \implies \: \left | \begin{array}{ccc} 1 & 1 & 1 \\ 1 & \omega & {\omega}^{2} \\ 1 & {\omega}^{2} & \omega\end{array} \right | \: = 3k \\ \\ \rm [ \because \: 1 + \omega + { \omega}^{2} = 0 \: and \: \: { \omega}^{7} = { ({ \omega}^{3} )}^{2}. \omega \: = \omega ]$

$\rm \: On \: applying \: \: \sf\red{ R_1 \longrightarrow R_1 + R_2 + R_3,} \: We \: get$

$\rm \left | \begin{array}{ccc} 3 & 1 + \omega + { \omega}^{2} & 1 + \omega + { \omega}^{2} \\ 1 & \omega & {\omega}^{2} \\ 1 & {\omega}^{2} & \omega \end{array} \right | \: = 3k$

$\implies \rm \left | \begin{array}{ccc} 3 & 0 & 0 \\ 1 & \omega & {\omega}^{2} \\ 1 & {\omega}^{2} & \omega \end{array} \right | \: = 3k$

$\rm\implies \: 3( { \omega}^{2} - { \omega}^{4} ) = 3k$

$\rm \implies \: ( { \omega}^{2} - \omega) = k$

$\rm \therefore \: k \: = \bigg( \dfrac{ - 1 - \sqrt{3} \: i}{2} \bigg) - \bigg( \dfrac{ - 1 + \sqrt{3} \: i }{2} \bigg)$

$\rm \implies \: k = \sqrt{3} \: i$

$\boxed{ \boxed{ \rm k = - z}}$

Answer 2

Answer:

$\omega$ be a complex number such that $2 \omega + 1 = z$ , where $z = \sqrt{-3}. If \ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ [tex] \left | \begin{array}{ccc} 1 & 1 & 1 \\ 1 & - {\omega}^{2} -1 & {\omega}^{2} \\ 1 & {\omega}^{2} & {\omega}^{7} \end{array} \right |$