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If A, B, C are angles of a triangle, then the value of [latex]\begin{vmatrix} e^{2iA} &e^{-iC} &e^{-iB} \\ e^{-iC} &e^{2iB} &e^{-iA} \\ e^{-iB}& e^{-iA} & e^{2iC} \end{vmatrix}[/latex] is

The value of the above determinant is calculated as follows.\(\begin{array}{l}\mathrm{A}+\mathrm{B}+\mathrm{C}=\pi and \mathrm{e}^{\mathrm{i\pi}}=\cos \pi+\mathrm{i} \sin \pi=-1\\ \mathrm{e}^{-i(\mathrm{~B}+\mathrm{C})}=\mathrm{e}^{-\mathrm{i}(\pi-\mathrm{A})}=-\mathrm{e}^{\mathrm{iA}} and \mathrm{e}^{-\mathrm{i}(\mathrm{B}+\mathrm{C})}=-\mathrm{e}^{\mathrm{i} \mathrm{A}}\\ \text... View Article