Question

If $\omega$ is a complex cube root of unity, then the matrix $A = \begin{bmatrix}1&\omega^{2}&\omega\\\ \omega ^{2}&\omega&1\\\ \omega&1&\omega ^{2}\end{bmatrix}$ is

• A. symmetric matrix
• B. diagonal matrix
• C. skew-symmetric matrix
• D. none of these

Answer 1

Answer: (A)

symmetric matrix

Answer 2

$A = \begin{bmatrix}1&\omega^{2}&\omega\\\ \omega ^{2}&\omega&1\\\ \omega&1&\omega ^{2}\end{bmatrix} \Rightarrow A^{T}= \begin{bmatrix}1&\omega ^{2}&\omega \\\ \omega ^{2}&\omega &1\\\ \omega &1&\omega ^{2}\end{bmatrix} = A$ $\because A^{T} = A$. Hence, $A$ is symmetric matrix.