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.