I recently saw the following problem on an entrance exam:

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 have no idea how to deal with the determinant of a 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 $\pm 1$. Hope someone could help me with this one. Thanks!