symmetric posititve definite matrix

Question

I am trying to find all values for $\alpha$ and $\beta$ for which

$$ A(\alpha, \beta)= \left[ \begin{matrix} 3&0&-2\\\alpha&3&2\\-2&2&\beta \end{matrix} \right] $$

is symmetric positive definite.

I know that $A$ is symmetric if $A=A^T$, but how do I show that it is symmetric positive definite?

Answer 1

Symmetry tells you what $\alpha$ should be.

Sylvester's criterion is one way to check positive-definiteness, which will help you find what values of $\beta$ are valid.

Answer 2

Compute the eigenvalues and choose $\beta$ that makes them all positive.