They're expecting you to know the spectral theorem for Hermitian matrices. Since M_j is obtained by striking out the jth row and jth column, it too is Hermitia

They're expecting you to know the spectral theorem for Hermitian matrices. Since M_j is obtained by striking out the jth row and jth column, it too is Hermitian, so it has has n-1 real eigenvalues (possibly with multiplicity). The notation $\lambda_i(M)$ ought to refer to the $i$th eigenvalue of M in sorted order to be unambiguous, but this does not matter for the proof.