If A and A^-1 have integer entries, why are detA and detA^-1 both ±1?

[tatianaiistb]

Homework Statement

If the entries of A and A^-1 are all integers, how do you know that both determinants are 1 or -1?

Homework Equations

The Attempt at a Solution

I know that
1 = det I = detAA^-1=detA * detA^-1= detA*(1/detA) = 1
Not sure how we get to - or the role integers play in it.

Thanks for your help!

 

[CompuChip]

So you have just shown that, if A = A^-1, then det A = 1 / det A.
Now consider the values that det A can take if all entries of A are integers.

 

[tatianaiistb]

Thanks! I was making the problem harder than it is!