On generalized eigenvalues of MAX matrices to MIN matrices and of LCM matrices to GCD matrices

We determine, for any n $\ge$ 1, the generalized eigenvalues of an n x n MAX matrix to the corresponding MIN matrix. We also show that a similar result holds for the generalized eigenvalues of an nxn LCM matrix to the corresponding GCD matrix when n $\le$ 4, but breaks down for n > 4. In addition, we prove Cauchy’s interlacing theorem for generalized eigenvalues, and we conjecture an unexpected connection between the OEIS sequence A004754 and the appearance of -1 as a generalized eigenvalue in the LCM-GCD setting.