First slide

Algebra of complex numbers

Question

Let $ω = - \frac{1}{2} + \frac{\sqrt{3}}{2} i$ Then the value of the determinant $Δ = |\begin{array}{ccc} 1 & 1 & 1 \\ 1 & - 1 - ω^{2} & ω^{2} \\ 1 & ω^{2} & ω^{4} \end{array}|$ is

Moderate

A

$3 ω$
B

$3 ω (ω - 1)$
C

$3 ω^{2}$
D

$3 ω (1 - ω)$

Solution

Using $1 + ω^{2} = - ω, ω^{4} = ω$ and applying
$C_{2} \to C_{2} - C_{1}, C_{3} \to C_{3} - C_{1}$ we get
$\begin{array}{l} Δ = |\begin{array}{ccc} 1 & 0 & 0 \\ 1 & ω - 1 & ω^{2} - 1 \\ 1 & ω^{2} - 1 & ω - 1 \end{array}| \\ = (ω - 1)^{2} - {(ω^{2} - 1)}^{2} \\ = (ω + ω^{2} - 2) (ω - 1 - ω^{2} + 1) \\ = (- 3) (ω - ω^{2}) = 3 ω (ω - 1) \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App