Let omega neq 1 be a cube root of unity and S | vedclass

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(A) Let the matrix be $M = \begin{bmatrix} 1 & a & b \ \omega & 1 & c \ \omega^2 & \omega & 1 \end{bmatrix}$.For the matrix to be non-singular,its d

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$2$

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(A) Let the matrix be $M = \begin{bmatrix} 1 & a & b \ \omega & 1 & c \ \omega^2 & \omega & 1 \end{bmatrix}$.For the matrix to be non-singular,its determinant must be non-zero,i.e.,$|M| 
eq 0$.$|M| = 1(1 - c\omega) - a(\omega - c\omega^2) + b(\omega^2 - \omega^2) = 1 - c\omega - a\omega + ac\omega^2$.$|M| = 1 - c\omega - a\omega + ac\omega^2 = (1 - a\omega)(1 - c\omega)$.For $|M| 
eq 0$,we must have $1 - a\omega 
eq 0$ and $1 - c\omega 
eq 0$.This implies $a\omega 
eq 1$ and $c\omega 
eq 1$.Since $\omega^3 = 1$,we have $\frac{1}{\omega} = \omega^2$. Thus,$a 
eq \omega^2$ and $c 
eq \omega^2$.Given that $a, b, c \in \{\omega, \omega^2\}$,the condition $a 
eq \omega^2$ implies $a = \omega$,and $c 
eq \omega^2$ implies $c = \omega$.However,$b$ can be either $\omega$ or $\omega^2$ because $b$ does not appear in the determinant expression.Thus,the possible values for $(a, b, c)$ are $(\omega, \omega, \omega)$ and $(\omega, \omega^2, \omega)$.Therefore,the number of distinct non-singular matrices is $2$.

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