I am trying to solve an exercise to prove if $A$ is semi-positive definite matrix, then its adjugate matrix $A^{*}$ is also semi-positive definite.
The proof comes to that, if $A$ is not full rank (the full rank case is trivial), then rank of $A^{*}$ is less or equal to 1, and then the characteristic polynomial is: $$|\lambda I - A^{*}|=\lambda^{n} - (A_{11}+A_{22}+...+A_{nn})\lambda^{n-1}$$ And the eigenvalue would be zero or positive, and thus finish the proof.
I wonder why this is true? How can I get the formula for the characteristic polynomial of the adjugate matrix?