Question

If $e^{i\theta }=\cos \theta +i\sin \theta$, find the value of $-\left|\begin{array}{ccc}1& e^{i\pi /3}& e^{i\pi /4}\\ e^{-i\pi /3}& 1& e^{i2\pi /3}\\ e^{-i\pi /4}& e^{-i2\pi /3}& 1\end{array}\right|-2^{\frac{1}{2}}$

Answer 1

Let $\Delta =\left|\begin{array}{ccc}1& e^{i\pi /3}& e^{i\pi /4}\\ e^{-i\pi /3}& 1& e^{i2\pi /3}\\ e^{-i\pi /4}& e^{-i2\pi /3}& 1\end{array}\right|$
Expanding along first column we get,
$\Delta =\left(1\right)\left|\begin{array}{cc}1& e^{i2\pi /3}\\ e^{-i2\pi /3}& 1\end{array}\right|-e^{-i\pi /3}\left|\begin{array}{cc}{e}^{i\pi /3}& e^{i\pi /4}\\ e^{-i2\pi /3}& 1\end{array}\right|+e^{-i\pi /4}\left|\begin{array}{cc}{e}^{i\pi /3}& e^{i\pi /4}\\ 1& e^{i2\pi /3}\end{array}\right|$
$\Rightarrow \Delta =1(1-1)-e^{-i\pi /3}(e^{i\pi /3}-e^{-5i\pi /12})+e^{-i\pi /4}(e^{\pi }-e^{i\pi /4})$
$\Rightarrow \Delta =-2+e^{3i\pi /4}+e^{-3i\pi /4}=-2+2\cos \left(3\pi /4\right)=-2-\sqrt{2}$
Hence required value is 2.