Let $A$ be an $n$-by-$n$ matrix with integer entries whose absolute values are bounded by a constant $C$. It is well-known that the entries of the inverse $A^{-1}$ can grow exponentially on $n$. (See the replies to https://math.stackexchange.com/questions/1146929/estimations-for-the-size-of-the-biggest-entry-in-an-inverse-matrix .) 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 $(\sqrt{n}C)^n$. $\endgroup$
    – user35593
    Mar 25 '19 at 16:08

Christian Remling's answer to this question: Smallest non-zero eigenvalue of a (0,1) matrix indicates that the answer is that the entry can grow super-exponentially.


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