Another one for the "yes" column. If you can diagonalize a matrix (such that all the entries are on the diagonal and all the off-diagonal entries are zero) using "legal" row operations, then the original matrix and diagonalized matrix are "similar", meaning they have the same eigenvalues. Since you can find the eigenvalues of a square matrix A by solving the equation det(A - jI) = 0 for j1 ... jN, which are the N eigenvalues of the NxN matrix A, it should be easy to see that the eigenvalues are equal in magnitude and sign to the diagonal elements of A if A is a diagonal matrix.
Hope this helps. Linear algebra rules.
R