True or False: If $A$ is square and $|\det(A)|=1$, then $A^{-1}$ has integer entries

Question

True or False: If $A$ is square and $|\det(A)|=1$, then $A^{-1}$ has integer entries.

I spent a good amount of time thinking about this problem. I think the answer would be false as I could see a possible counterexample. Could somebody please help me out? Thank you in advance.

Answer 1

Hint: (I'm assuming $A$'s entries are integers.) Easy to see on a $2\times 2$ matrix. In general:

$$ A^{-1} = \det(A)^{-1} \mathrm{adj}(A) $$

(adjugate matrix description here)