Question is in regards to 6 (with reference to 4).
I get the hint, but am not seeing how some principal matrix being non-singular helps to prove rank of A-rI is n-1.
What you have to show is that if is a matrix which is not invertible and such that a principal submatrix of size is invertible then the rank of is . After permuting if necessary the rows and the columns, you can assume that it's the submatrix after deleting the -th row and -th column. Let the submatrix, and consider the blockwise matrix . Multipliying by , we can see that the rank of is at least . It can't be more since is not invertible.
I understand what I'm meant to show; not sure I follow your proof to be honest (syntax is a bit awkward).
Jumped on Wiki and "determinantal rank" jumped out; that seems like the easiest route with n-1 being largest possible rank of principle submatrix.