Let $a(M)$ be the maximum absolute value of entries of matrix $M\in\mathsf{GL}\_k(\mathbb Z)$.


$M^{-1}\in\mathsf{GL}\_k(\mathbb Z)$ holds.



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> What is a good upper bound for $|a(M)-a(M^{-1})|$?
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I am thinking whether the dependence could be a little smaller than fully exponential in $k$ for $a(M)\cdot a(M^{-1})$ which will reflect upper bound for $|a(M)-a(M^{-1})|$.



Upper bounds for difference of entries between matrices and their inverses in $\mathsf{GL}_k(\mathbb Z)$
For $k=2$, the upper bound is zero. 


For $k>2$, there is no upper bound. E.g., let $$M=\pmatrix{1&1&1\cr9&10&11\cr-n&n&3n-1\cr}$$ Then $$M^{-1}=\pmatrix{10-19n&2n-1&-1\cr38n-9&1-4n&2\cr-19n&2n&-1\cr}$$ and $a(M)-a(M^{-1})=(3n-1)-(38n-9)=-(35n-8)$ is unbounded. 


