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 $\displaystyle B$ is a $\displaystyle n\times n$ matrix which is not invertible and such that a principal submatrix of size $\displaystyle (n-1)\times (n-1)$ is invertible then the rank of $\displaystyle B$ is $\displaystyle n-1$. After permuting if necessary the rows and the columns, you can assume that it's the submatrix after deleting the $\displaystyle n$-th row and $\displaystyle n$-th column. Let $\displaystyle B'$ the submatrix, and consider the blockwise matrix $\displaystyle C=\begin{pmatrix}B'&0\\ 0&1\end{pmatrix}$. Multipliying by $\displaystyle C$, we can see that the rank of $\displaystyle B$ is at least $\displaystyle n-1$. It can't be more since $\displaystyle B$ 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.