# A sufficient condition for a real symmetric matrix to be positive definite

I was recently asked to show this for matrix theory class:

Let A be an $n \times n$ real symmetric matrix such that $\det(A)\geq 0$ with n-1 nested principal (not necessarily leading) minors which are positive. We are asked to prove that A is positive semidefinite.

To be honest I know the characterization theorem on positive semidefinite matrices but I cannot see how this leads to an equivalent condition of positive semi definiteness? Maybe the limit of positive semi definite matrices? I therefore ask here, and I thank all helpers.