# Math Help - Matrix Analysis - Horn Question

1. ## Matrix Analysis - Horn Question

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.

2. ## Re: Matrix Analysis - Horn Question

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

3. ## Re: Matrix Analysis - Horn Question

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.