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Question: If \(\omega\) is an imaginary root of unity, then the value of \(\left| \begin{matrix} a & b\omega^{...

If ω\omega is an imaginary root of unity, then the value of ∣abω2aωbωcbω2cω2aωc∣\left| \begin{matrix} a & b\omega^{2} & a\omega \\ b\omega & c & b\omega^{2} \\ c\omega^{2} & a\omega & c \end{matrix} \right| is.

A

a3+b3+c3−3abca^{3} + b^{3} + c^{3} - 3abc

B

a2b−b2ca^{2}b - b^{2}c

C

0

D

a2+b2+c2a^{2} + b^{2} + c^{2}

Answer

0

Explanation

Solution

We have ∣abω2aωbωcbω2cω2aωc∣\left| \begin{matrix} a & b\omega^{2} & a\omega \\ b\omega & c & b\omega^{2} \\ c\omega^{2} & a\omega & c \end{matrix} \right|

= ∣a(1+ω)bω2aωb(ω+ω2)cbω2c(ω2+1)aωc∣\left| \begin{matrix} a(1 + \omega) & b\omega^{2} & a\omega \\ b(\omega + \omega^{2}) & c & b\omega^{2} \\ c(\omega^{2} + 1) & a\omega & c \end{matrix} \right| , {C1→C1+C3}\{ C_{1} \rightarrow C_{1} + C_{3}\}

$\left| \begin{matrix}

  • a\omega^{2} & b\omega^{2} & a\omega \
  • b & c & b\omega^{2} \
  • c\omega & a\omega & c \end{matrix} \right| = \omega^{2}\omega\left| \begin{matrix}
  • a & b & a\omega^{2} \
  • b & c & b\omega^{2} \
  • c & a & c\omega^{2} \end{matrix} \right|$

= $\omega^{2}\left| \begin{matrix}

  • a & b & a \
  • b & c & b \
  • c & a & c \end{matrix} \right|== - \omega^{2}\left| \begin{matrix} a & b & a \ b & c & b \ c & a & c \end{matrix} \right| = 0$.