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.

Are the entries of $A$ integers? What are your thoughts on the possible counterexample? — robjohn, Jul 05 '20 at 23:49
a counterexample could be (1/3, 1) for row 2 and (1,9) for row 2 det(A)=1 A^-1 : (9, -1) for row one and (-2, 1/3) for row 2 — Nick, Jul 06 '20 at 03:07
Does this answer your question? When the inverse of a matrix with integer entries also have integer entries — PinkyWay, Jul 06 '20 at 07:39

ir7 · Answer 1 · 2020-07-06T00:33:13.607

2

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)

edited Jul 06 '20 at 00:33

answered Jul 06 '20 at 00:27

ir7

6,249

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

1 Answers1