Having a positive determinant is one of the main properties of a Postive Semi-Definite matrix. How do I prove it?
let $\displaystyle \lambda$ be an eigenvalue of $\displaystyle A.$ then $\displaystyle A\bold{x}=\lambda \bold{x},$ for some $\displaystyle \bold{x} \neq \bold{0}.$ thus $\displaystyle 0 \leq \bold{x}^* A \bold{x}=\lambda \bold{x}^* \bold{x}=\lambda ||\bold{x}||^2$ and so $\displaystyle \lambda \geq 0.$ the result now follows because $\displaystyle \det A$ is the product of the eigenvalues of $\displaystyle A.$