If ω is a complex cube root of unity, then the matrix A=1ω2ωω2ω1ω1ω2 is a

# If $\omega$ is a complex cube root of unity, then the matrix $A=\left[\begin{array}{ccc}1& {\omega }^{2}& \omega \\ {\omega }^{2}& \omega & 1\\ \omega & 1& {\omega }^{2}\end{array}\right]$ is a

1. A

singular matrix

2. B

non-singular matrix

3. C

skew-symmetric matrix

4. D

none of these

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### Solution:

We have

$|A|=\left|\begin{array}{ccc}1& {\omega }^{2}& \omega \\ {\omega }^{2}& \omega & 1\\ \omega & 1& {\omega }^{2}\end{array}\right|=\left|\begin{array}{ccc}1+{\omega }^{2}+\omega & {\omega }^{2}& \omega \\ {\omega }^{2}+\omega +1& \omega & 1\\ \omega +1+{\omega }^{2}& 1& {\omega }^{2}\end{array}\right|=0$

using ${C}_{1}\to {C}_{1}+{C}_{2}+{C}_{3}$ and $1+\omega +{\omega }^{2}=0$

$\therefore$A is a singular matrix.

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