Answer
\begin{align*}
A^{-1} & =\frac{\text{adj}\,A}{\det A}=\text{adj}\,A
\\ \Rightarrow
(A^{-1})_{ij} & =
C_{ji}=
(-1)^{i+j}\det A_{ji}
\end{align*}
Because $\det A_{ji}$ are obtained by multiplications, additions, and subtractions between the entries of $A$, they are all integers
$\Rightarrow$ all the entries in $A^{-1}$ are integers
Work Step by Step
\begin{align*}
A^{-1} & =\frac{\text{adj}\,A}{\det A}=\text{adj}\,A
\\ \Rightarrow
(A^{-1})_{ij} & =
C_{ji}=
(-1)^{i+j}\det A_{ji}
\end{align*}
Because $\det A_{ji}$ are obtained by multiplications, additions, and subtractions between the entries of $A$, they are all integers
$\Rightarrow$ all the entries in $A^{-1}$ are integers
