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

• A. 4
• B. -4
• C. -8
• D. 8

Answer 1

The correct option is B -4

$D=\left|\begin{array}{ccc}{e}^{2iA}& e^{iC}& e^{iB}\\ e^{iC}& e^{-2iB}& e^{iA}\\ e^{iB}& e^{-iA}& e^{-2iC}\end{array}\right|$
$=e^{-iA}{e}^{-iB}{e}^{-iC}\left|\begin{array}{ccc}{e}^{-iA}& e^{i(A+C)}& e^{i(A+B)}\\ e^{i(B+C)}& e^{iB}& e^{i(A+B)}\\ e^{i(B+C)}& e^{i(A+C)}& e^{-iC}\end{array}\right|$
As $A+B+C=\pi$
So $A+C=\pi -B,B+C=\pi -A,A+B=\pi -C$
$D=e^{i\pi }\left|\begin{array}{ccc}{e}^{-iA}& e^{i(\pi -B)}& e^{i(\pi -C)}\\ e^{i(\pi -A)}& e^{iB}& e^{i(\pi -C)}\\ e^{i(\pi -A)}& e^{i(\pi -B)}& e^{iC}\end{array}\right|$
$D=-\left|\begin{array}{ccc}{e}^{iA}& -e^{-iB}& -e^{-C}\\ -e^{iA}& e^{-iB}& -e^{iC}\\ -e^{-iA}& -e^{-iB}& e^{-iC}\end{array}\right|$
$=-e^{-iA}{e}^{-iB}{e}^{-iC}\left|\begin{array}{ccc}1& -1& -1\\ -1& 1& -1\\ -1& -1& 1\end{array}\right|=-e^{-i\pi }(-4)=-4$