First slide

Introduction to Determinants

Question

If A, B, C are angles of a triangle, then the value of $|\begin{array}{lll} e^{2 iA} e^{- iC} e^{- iB} \\ e^{- iC} e^{2 iB} e^{- iA} \\ e^{- iB} e^{- iA} e^{2 iC} \end{array}|$ is

Difficult

A

1
B

-1
C

-2
D

-4

Solution

Since $A + B + C = π and e^{iπ} = \cos π + isin π = - 1$ ,
$e^{i (B + C)} = e^{i (π - A)} = - e^{- iA} and e^{- i (B + C)} = - e^{iA}$
By taking $e^{iA}, e^{iB}, e^{iC}$ common from $R_{1}, R_{2} and R_{3}$ , respectively,
we have
$\begin{matrix} Δ = - |\begin{array}{ccc} e^{iA} & e^{- i (A + C)} & e^{- i (A + B)} \\ e^{- i (B + C)} & e^{iB} & e^{- i (A + B)} \\ e^{- i (B + C)} & e^{- i (A + C)} & e^{iC} \end{array}| \\ = - |\begin{array}{ccc} e^{iA} & - e^{iB} & - e^{iC} \\ - e^{iA} & e^{iB} & - e^{iC} \\ - e^{iA} & - e^{iB} & e^{iC} \end{array}| \end{matrix}$
By taking $e^{iA}, e^{iB}, e^{iC}$ common from C₁, C₂ and C₃, respectively,
we have
$Δ = |\begin{array}{ccc} 1 & - 1 & - 1 \\ - 1 & 1 & - 1 \\ - 1 & - 1 & 1 \end{array}| = - 4$

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