I have this problem that I really can't solve.

For which s and t do A and B have all eigenvalues λ > 0 (therefore positive definite)?

\(\displaystyle A=\begin{pmatrix}

s & -4 & -4 \\ -4 & s & -4 \\ -4 & -4 & s

\end{pmatrix}\) and \(\displaystyle B=\begin{pmatrix}

t & 3 & 0 \\ 3 & t & 4 \\ 0 & 4 & t

\end{pmatrix}\)

I tried to use the characteristic polynomial but I can't solve it since it's third degree. I also tried Cholesky decomposition for symmetric matrices, but the elements of the matrices are too complicated to be used in a useful way.

Could someone help me?