If A, B, C are angles of a triangle, then the value of e2iA e−iC e−iBe−iC e2iB e−iAe−iB e−iA e2iC is
1
-1
-2
-4
Since A+B+C=π and eiπ=cosπ+isinπ=−1,
ei(B+C)=ei(π−A)=−e−iA and e−i(B+C)=−eiA
By taking eiA, eiB, eiC common from R1, R2 and R3, respectively,
we have
Δ=−eiAe−i(A+C)e−i(A+B)e−i(B+C)eiBe−i(A+B)e−i(B+C)e−i(A+C)eiC=−eiA−eiB−eiC−eiAeiB−eiC−eiA−eiBeiC
By taking eiA, eiB, eiC common from C1, C2 and C3, respectively,
Δ=1−1−1−11−1−1−11=−4