Help solving a couple of questions with Linear Algebra

Hi there guys,
I have tried and tried and tried. I can't seem to get anywhere with these questions, Just wondering if anybody here could help me out!Attachment 26451
Any help would be great! Please and thank you!

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:
$\displaystyle t^2-tr(A)t+det(A)=0$, 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.