Let $A$ be an $n$by$n$ matrix with integer entries whose absolute values are bounded by a constant $C$. It is wellknown that the entries of the inverse $A^{1}$ can grow exponentially on $n$. (See the replies to https://math.stackexchange.com/questions/1146929/estimationsforthesizeofthebiggestentryinaninversematrix .) Can they grow more rapidly than exponentially? Or are they bounded by $(C+O(1))^n$?

$\begingroup$ If it would be possibel to add a row and column to a Hadamard (e.g. Walsh matrices)$ such that determinant is 1 the entries could grow like $(\sqrt{n}C)^n$. $\endgroup$– user35593Mar 25 '19 at 16:08
Christian Remling's answer to this question: Smallest nonzero eigenvalue of a (0,1) matrix indicates that the answer is that the entry can grow superexponentially.