Question

If w is a complex cube root of unity, then the matrix A =

$\begin{bmatrix} 1 & w^{2} & w \\ w^{2} & w & 1 \\ w & 1 & w^{2} \end{bmatrix}$is a-

• A. Singular matrix
• B. Non-singular matrix
• C. Skew symmetric matrix
• D. None of these

Answer 1

Answer: (A)

Singular matrix

Answer 2

We have

|A|=$\begin{bmatrix} 1 & w^{2} & w \\ w^{2} & w & 1 \\ w & 1 & w^{2} \end{bmatrix}$=$\begin{bmatrix} 1 + w^{2} + w & w^{2} & w \\ w^{2} + w + 1 & w & 1 \\ w + 1 + w^{2} & 1 & w^{2} \end{bmatrix}$

[Using C₁ ® C₁ + C₂ + C₃]

= $\begin{bmatrix} 0 & w^{2} & w \\ 0 & w & 1 \\ 0 & 1 & w^{2} \end{bmatrix}$ = 0

\ A is a singular matrix.