The sum (product) of the eigenvalues of a matrix is a trace (determinant) of a matrix. One can find the proof at the following link on page 3,
https://www.adelaide.edu.au/mathslearning/play/seminars/evalue-magic-tricks-handout.pdf
The matrice $A$ and $A^{T}$ have the same set of eigenvalues, a proof can be found at the same link on page 2.
PS. The sum and product are done taking multiplicity into account, as can be seen in a proof provided by the link.
https://www.adelaide.edu.au/mathslearning/play/seminars/evalue-magic-tricks-handout.pdf
The matrice $A$ and $A^{T}$ have the same set of eigenvalues, a proof can be found at the same link on page 2.
PS. The sum and product are done taking multiplicity into account, as can be seen in a proof provided by the link.
