Lately I saw a problem of entrance exam as following:

Let $A$ be a square matrix. Prove that there is a diagonal matrix $D$ whose diagonal entries are either $+1$ or $-1$ such that $det(A+D)\neq 0$.

I totally have no idea how to deal with determinant of sum of matrices. I think it can be proved by contradiction. But I don't see what would lead to if $det(A+D)=0$ for all diagonal matrix $D$ with diagonal entries $\pm1$.

Hope someone could help me with this one, thanks!