I think you neglected to mention that A is a 2 by 2 matrix! For any larger matrix, your result is generally false.
For a 2 by 2 matrix, the characteristic equation is:
$\displaystyle t^2-tr(A)t+det(A)=0$, and so Cayley Hamilton says $\displaystyle A^2-tr(A)A+det(A)I=0$
So $\displaystyle I={tr(A)A-A^2\over tr(A)}$ or $\displaystyle I=AB$ with $\displaystyle B={tr(A)I-A\over tr(A)}$
Don't you know then that this implies $\displaystyle B=A^{-1}$?
The remaining questions should be easy using this form for A inverse.