${\mathbf{x}}^{\prime}=A\mathbf{x}$

where

$A=\left[\begin{array}{ccc}a& 0& 4\\ -1& -1& 0\\ -2-a& 0& -3\end{array}\right]$

In what interval of a is the system asymptotically stable, and for what value of a is the system stable/unstable if such a case even exists?

Attempt

For a system to be asymptotically stable the real part of all eigenvalues must be negative.

$\mathrm{\Re}(\lambda )<0,\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\text{for all}\phantom{\rule{mediummathspace}{0ex}}\phantom{\rule{mediummathspace}{0ex}}\lambda $

Since we are dealing with a $3\times 3$ matrix, I used maple to find the eigenvalues from the system matrix. I also tried to use Maple to solve the inequality case for the eigenvalues, but I don't think I'm getting correct results.

The results I get from Maple state that the system is asymptotically stable when

$-8<a\le 5-4\sqrt{3}$

I spoke with my peers, and they said that this interval of a is wrong. Can anyone see what is going wrong?