1. Independently of $\displaystyle a,b$ we have $\displaystyle a_{11},a_{12},a_{13}$ non negative.
2. $\displaystyle a_{22},a_{23},a_{33}$ are non negative iff $\displaystyle 4b-1<0$ i.e. $\displaystyle b<1/4$ .
3. For $\displaystyle b<1/4$ , $\displaystyle a_{32}$ is non negative iff $\displaystyle b\geq 0$ .
4. For $\displaystyle 0\leq b<1/4$ , $\displaystyle a_{21}$ is non negative iff $\displaystyle 8a+1\geq 0$ i.e. $\displaystyle a\geq -1/8$ .
5. For $\displaystyle 0\geq b<1/4$ and $\displaystyle a\geq -1/8$ , $\displaystyle a_{31}$ is non negative iff $\displaystyle b+2a\geq 0$ .
So, the solution is the region $\displaystyle R$ of the $\displaystyle ab$ plane:
$\displaystyle R \equiv\begin{Bmatrix}2a+b\geq 0\\a\geq -1/8\\0\leq b<1/4\end{matrix}$