let $A$ be a symmetric $n$ by $n$ matrix over $\mathbb{GF}(2)$. Using elementary linear algebra, it is quite easy to show that diag $A$ is in the range of $A$, where diag $A = [a_{11},a_{22}, \dots, a_{nn}]$. You can find a proof here. The link also provides a proof that is longer, but does not use theorems from linear algebra.
As a matter of fact, the problem that I encountered first was a special case, and so far I've only been able to prove the general case.
I'm searching for a solution for the special case where diag $A = [1, 1, \dots, 1]$, which does not use theorems from linear algebra, and is simpler/shorter than the proof suggested in the link above.