First slide

Introduction to Determinants

Question

Let $x = \cos \frac{π}{3} + i \sin \frac{π}{3}$ and $Δ = |\begin{array}{ccc} 1 & x & x^{2} \\ x^{3} & 1 & x \\ 1 & x^{2} & 1 \end{array}|$ then numerical value of $∆$ is

Moderate

A

0
B

-1
C

8
D

-4

Solution

Use $C_{3} \to C_{3} - x C_{2}$ and $C_{2} \to C_{2} - x C_{1}$
$Δ = |\begin{array}{ccc} 1 & 0 & 0 \\ x^{2} & 1 - x^{3} & 0 \\ 1 & x^{2} - x & 1 - x^{3} \end{array}|$
$= {(1 - x^{3})}^{2} = [1 - \cos π - i \sin π]^{2} = 8$
$[De Moivre's Theorem]$

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