Prove there exists vector $y\in \mathbb{Z}_2^n \ s.t. Ay=(a_{11},a_{22},\dots,a_{nn})^t.$

Question

Let $A\in\text{Mat}_n(\mathbb{Z}_2)$ be a symmetric matrix. Prove there exists vector $y\in \mathbb{Z}_2^n \ s.t. Ay=(a_{11},a_{22},\dots,a_{nn})^t.$

So I tried some numerical examples and playing with the system of linear equations that comes out of the multiplication, but couldn't find a generalized solution.

Any help appreciated.