The matrix A, Being the general one
[a,b
c,d]
Where Tr(A) = a+d and Det(A) is ad-bc
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:
, and so Cayley Hamilton says
So or with
Don't you know then that this implies ?
The remaining questions should be easy using this form for A inverse.