Symmetric Matrices & Quadratic Forms · Hard problem

A matrix $A = \begin{pmatrix}-1&0&0\\0&5&4\\0&4&5\end{pmatrix}$ has a positive $(2,2)$ entry, and its lower-right $2\times2$ block $\begin{pmatrix}5&4\\4&5\end{pmatrix}$ has positive determinant ($25-16=9$) and positive trace — yet $A$ is NOT positive definite, because positive-definiteness requires checking the LEADING principal minors (starting from the top-left corner), not just any block. Find the smallest eigenvalue of $A$ to confirm this.