I have a small question regarding the adjugate matrix.
Suppose we have a square ($n \times n$) singular matrix with a rank of $n-1$.
Now I have two questions I'm trying to investigate:
Is it possible that the adjugate matrix rank won't be changed? That is, if we have a $n \times n$ matrix with a rank of $n-1$ the adjugate will have the same rank ($n-1$).
I know that in this case (rank of A is $n-1$) that $\mathrm{adj}(\mathrm{adj}(A))$ is $0$. I don't understand why. Is there any relation between these two questions?
Thanks alot, Guy