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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.

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Relabel the rows and columns so that the nested principal minors are the leading ones. Apply matrix congruence to kill off all but the last entries on the last row/column. The rest is obvious.

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