Prove that a $3\times3$
is positive definite if and only if $a > 0$, $ad - b^2 > 0 $ , $\det K > 0$. Also prove that an $n\times n$ matrix $K > 0$ is positive definite if and only if all the upper left square $k \times k$ subdeterminants are positive for $k = 1, \ldots, n$.
For the first part I know that in order it to be a positive definite then the quadratic function $q(x)$ can be transformed by completing the squares but I don't know how to prove $a > 0$, $ad - b^2 > 0 $ , $\det K > 0$. For the second part I have no idea how to do.