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

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

A
singular matrix
B
non-singular matrix
C
skew-symmetric matrix
D
none of these

Solution:

We have

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

using $C_{1} \to C_{1} + C_{2} + C_{3}$ and $1 + ω + ω^{2} = 0$

$∴$ A is a singular matrix.

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