First slide

Algebra of complex numbers

Question

Suppose z₁, z₂, z₃ are three complex numbers, and $Δ = \frac{1}{4 i} |\begin{array}{ccc} 1 & z_{1} & {\bar{z}}_{1} \\ 1 & z_{2} & {\bar{z}}_{2} \\ 1 & z_{2} & {\bar{z}}_{3} \end{array}|$ then

Moderate

A

$R e (∆) = 0$
B

$I m (∆) = 0$
C

$R e (∆) \geq 0$
D

$I m (∆) \leq 0$

Solution

$\bar{Δ} = \frac{1}{- 4 i} |\begin{array}{ccc} 1 & {\bar{z}}_{1} & z_{1} \\ 1 & {\bar{z}}_{2} & z_{2} \\ 1 & {\bar{z}}_{3} & z_{3} \end{array}| = Δ$
$\Rightarrow ∆$ is purely real $\Rightarrow I m (∆) = 0$

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