Multiply by [tex]A^{-1}[/tex] on the right of both sides of [tex]A^3=I[/tex], then use the fact that matrix multiplication is associative.
If [tex]A^3=I[/tex], then
[tex]A^3A^{-1}=IA^{-1}[/tex]
[tex]A^2(AA^{-1})=A^{-1}[/tex]
[tex]A^2I=A^{-1}[/tex]
[tex]\implies A^2=A^{-1}[/tex]
