Matrix finds the cube

Algebra Level 4

Let \(A = \left( \begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix} \right)\) be a matrix where \(a, b, c\) are complex numbers. If \(abc = 1\) and \(A^T A = I\), what is the value of \(a^3 + b^3 + c^3\)?

Clarification: \(A^T\) is the transpose of \(A\). \(I\) is the identity matrix of order \(3 \times 3\).

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