Question

If A, B and C are the angle of a triangle $D=\left|\begin{array}{ccc}{e}^{-2iA}& e^{iC}& e^{iB}\\ e^{iC}& e^{-2iB}& e^{iA}\\ e^{iB}& e^{iA}& e^{-2iC}\end{array}\right|$ where $i=\sqrt{-1}$, then find the value of D.

Answer 1

$A+B+C=180$

$D=\left|\begin{array}{ccc}{e}^{-2iA}& e^{iC}& e^{iB}\\ e^{iC}& e^{-2iB}& e^{iA}\\ e^{iB}& e^{iA}& e^{-2iC}\end{array}\right|$

$=e^{-2iA}(e^{-2i(B+C)}-e^{2iA})-e^{iC}(e^{-iC}-e^{i(A+B)})+e^{iB}(e^{i(A+B)}-e^{-iB})$

$=e^{+2i\left(\Pi \right)}-1-1+e^{+2i\left(\Pi \right)}+e^{2i\left(\Pi \right)}-1$

$=3e^{2i\left(\Pi \right)}-3$

$=3(e^{2\Pi }-1)=3(\cos 2\Pi +i\sin 2\Pi -1)$

$=3(1-1)=0$

$\therefore D=0$ .