Linear Algebra – Prove A Inverse Contains Integer Entries if det(A) = ±1

Question

Prove that if $A$ is a square matrix with integer entries and $\det(A) = \pm 1$, then $A^{-1}$ contains all integer entries.

I'm really thrown off by this one, its unlike all the examples I've seen.. I just dont know where to start… how can I begin to prove this?

Answer 1

Hint: $A^{-1}=\frac{1}{\det(A)}\text{adj}(A)$.