[tex]A^2=2I[/tex]
Recall that [tex]I=AA^{-1}=A^{-1}A[/tex], so
[tex]A^2=2(A^{-1}A)[/tex]
Multiply on the right by [tex]A^{-1}[/tex] and regroup terms:
[tex]A^2A^{-1}=2(A^{-1}A)A^{-1}[/tex]
[tex]A(AA^{-1})=2A^{-1}(AA^{-1})[/tex]
[tex]A=2A^{-1}[/tex]
and finally solve for [tex]A^{-1}[/tex]:
[tex]\boxed{A^{-1}=\dfrac12A}[/tex]
