Suppose $A$ is a real, square matrix which is singular, and call the entries $a_{i j}$. Let $f$ be a real valued function and consider the matrix $A^f$ whose entries are $f(a_{i j})$. Are there any functions $f$ such that $A^f$ is not singular, even though $A$ is? If so, what kind of properties would the function have?
EDIT: Let me ask that $f$ be continuous, and I'm not looking for a function which can make any singular $A$ into a nonsingular $A^f$ (I assume this is not possible?). What are examples of specific pairs of singular $A$, continuous $f$, yielding nonsingular $A^f$?
Let $A = \left(\begin{array}{ss} 0 & 0\\ 0 & 1 \end{array}\right)$ and $f(x) = x+1$, then $A^f = \left(\begin{array}{ss} 1 & 1\\ 1 & 2 \end{array}\right)$ is non-singular.