I've been stuck on this for a while:
Show that a square matrix A has no inverse when each row of A sums to 0.
Any help would be appreciated!
An idea: induction on n = number of rows: for n= 1, 2 is almost immediate, so assume for n: now begin simplifying your matrix by Gauss making zeros on the first column (Note, or in fact: prove, that after this is done all the rows STILL sum to zero!) except perhaps the entry 1-1 and now develop by minors wrt the first column...
Tonio
$\displaystyle A\begin{bmatrix} 1 \\ 1 \\ \vdots \\ 1 \end{bmatrix}=\begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix}, span(\begin{bmatrix} 1 \\ 1 \\ \vdots \\ 1 \end{bmatrix}) \in \ker(A) \neq \phi \ ,\ \therefore A $ is noninvertible