First slide

Introduction to Determinants

Question

Let $ω = \frac{1}{2} (- 1 + \sqrt{3} i)$ and $Δ = |\begin{array}{ccc} 1 & 1 & 1 \\ 1 & - 1 - ω^{2} & ω^{2} \\ 1 & ω^{2} & ω^{4} \end{array}|$ then $Δ$ equals

Moderate

A

$3 ω$
B

$3 ω (ω - 1)$
C

$3 ω^{2}$
D

$3 ω (1 - ω)$

Solution

Using $ω^{3} = 1$ and $1 + ω + ω^{2} = 0$ , and applying $C_{1} \to C_{1} + C_{2} + C_{3},$ we get
$Δ = |\begin{array}{ccc} 3 & 1 & 1 \\ 0 & ω & ω^{2} \\ 0 & ω^{2} & ω \end{array}| = 3 (ω^{2} - ω) = 3 ω (ω - 1)$

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