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The maximum value of "a" such that the matrix $\begin{pmatrix}
-3&0&-2 \\
1&-1&0  \\
0&a&-2 
\end{pmatrix}$ has three linearly independent real eigenvectors is

  1. $\frac{2}{3\sqrt{3}}$
  2. $\frac{1}{3\sqrt{3}}$
  3. $\frac{1+2\sqrt{3}}{3\sqrt{3}}$
  4. $\frac{1+\sqrt{3}}{3\sqrt{3}}$
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