Suppose that the characteristic equation of \( M \in \mathbb{C}^{3 \times 3} \) is
\[
\lambda^3 + \alpha \lambda^2 + \beta \lambda - 1 = 0,
\]
where \( \alpha, \beta \in \mathbb{C} \) with \( \alpha + \beta \neq 0 \).
Which of the following statements is TRUE?