Discussion
Most generic eigenvalues of a 3 × 3 traceless matrix with zero determinant are given as (a is some constant) :
Answer: C
The generic eigenvalues are a, -a, 0.
For a \(3 \times 3\) matrix, the trace is the sum of the eigenvalues and the determinant is their product:
\[ \begin{aligned} \lambda_{1} + \lambda_{2} + \lambda_{3} &= 0 \\ \lambda_{1}\lambda_{2}\lambda_{3} &= 0 \end{aligned} \]
Zero determinant forces one eigenvalue to be 0; then the other two must sum to zero, so they are \(a\) and \(-a\). This is the most general form with one free constant.
0, 0, 0 is a special case (\(a = 0\)). 0, 0, \(a\) has trace \(a\), not 0, and \(a, -2a, a\) has determinant \(-2a^{3}\), not 0.
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