First slide

Introduction to Determinants

Question

If w is a complex cube root of unity, then value of $Δ = |\begin{array}{lll} a_{1} + b_{1} w a_{1} w^{2} + b_{1} c_{1} + b_{1} \bar{w} \\ a_{2} + b_{2} w a_{2} w^{2} + b_{2} c_{2} + b_{2} \bar{w} \\ a_{3} + b_{3} w a_{3} w^{2} + b_{3} c_{3} + b_{3} \bar{w} \end{array}|$ is

Moderate

A

0
B

-1
C

2
D

none of these

Solution

$Δ = |\begin{array}{lll} a_{1} + b_{1} w a_{1} w^{2} + b_{1} c_{1} + b_{1} \bar{w} \\ a_{2} + b_{2} w a_{2} w^{2} + b_{2} c_{2} + b_{2} \bar{w} \\ a_{3} + b_{3} w a_{3} w^{2} + b_{3} c_{3} + b_{3} \bar{w} \end{array}|$
Operating $C_{2} \to {wC}_{2}$ , we have
$\begin{matrix} Δ = \frac{1}{w} |\begin{array}{lll} a_{1} + b_{1} w & a_{1} w^{3} + b_{1} w & c_{1} + b_{1} \bar{w} \\ a_{2} + b_{2} w & a_{2} w^{3} + b_{2} w & c_{2} + b_{2} \bar{w} \\ a_{3} + b_{3} w & a_{3} w^{3} + b_{3} w & c_{3} + b_{3} \bar{w} \end{array}| \\ = \frac{1}{w} |\begin{array}{lll} a_{1} + b_{1} w & a_{1} + b_{1} w & c_{1} + b_{1} \bar{w} \\ a_{2} + b_{2} w & a_{2} + b_{2} w & c_{2} + b_{2} \bar{w} \\ a_{3} + b_{3} w & a_{3} + b_{3} w & c_{3} + b_{3} \bar{w} \end{array}| (∵ ω^{3} = 1) \\ = 0 \end{matrix}$

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