Question

Find the value of the given determinant. $\left|\begin{array}{ccc}1& \omega & 2{\omega }^2\\ 2& 2{\omega }^2& 4{\omega }^3\\ 3& 3{\omega }^3& 6{\omega }^4\end{array}\right|$

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is A

$0$

Using the formula for $3\times 3$ determinant find the value of the given determinant.

We have been given a determinant$\left|\begin{array}{ccc}1& \omega & 2{\omega }^2\\ 2& 2{\omega }^2& 4{\omega }^3\\ 3& 3{\omega }^3& 6{\omega }^4\end{array}\right|$.

We need to find the value of a given determinant.

Using the formula for $3\times 3$ determinant $\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|=a\times \left[\right(e\times i)-(h\times f\left)\right]-b\times \left[\right(d\times i)-(g\times f\left)\right]+c\times \left[\right(d\times h)-(g\times f\left)\right]$ we find the value of the given determinant.

$$\begin{gathered} \Rightarrow \left|\begin{array}{ccc}1& \omega & 2{\omega }^2\\ 2& 2{\omega }^2& 4{\omega }^3\\ 3& 3{\omega }^3& 6{\omega }^4\end{array}\right|=1\left[\right(2{\omega }^2\times 6{\omega }^4)-(3{\omega }^3\times 4{\omega }^3\left)\right]-\omega \left[\right(2\times 6{\omega }^4)-(3\times 4{\omega }^3\left)\right]+2{\omega }^2\left[\right(2\times 3{\omega }^3)-(3\times 2{\omega }^2\left)\right] \\ \Rightarrow \left|\begin{array}{ccc}1& \omega & 2{\omega }^2\\ 2& 2{\omega }^2& 4{\omega }^3\\ 3& 3{\omega }^3& 6{\omega }^4\end{array}\right|=(12{\omega }^6-12{\omega }^6)-\omega (12{\omega }^4-12{\omega }^3)+2{\omega }^2(6{\omega }^3-6{\omega }^2) \\ \Rightarrow \left|\begin{array}{ccc}1& \omega & 2{\omega }^2\\ 2& 2{\omega }^2& 4{\omega }^3\\ 3& 3{\omega }^3& 6{\omega }^4\end{array}\right|=0-12{\omega }^5+12{\omega }^4+12{\omega }^5-12{\omega }^4 \\ \Rightarrow \left|\begin{array}{ccc}1& \mathrm{\omega }& 2{\mathrm{\omega }}^2\\ 2& 2{\mathrm{\omega }}^2& 4{\mathrm{\omega }}^3\\ 3& 3{\mathrm{\omega }}^3& 6{\mathrm{\omega }}^4\end{array}\right|=\mathbf{0} \end{gathered}$$

Therefore, option (A) is the correct answer.