# Thread: Help solving a couple of questions with Linear Algebra

1. ## 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!

Any help would be great! Please and thank you!

2. ## Re: Help solving a couple of questions with Linear Algebra

The matrix A, Being the general one
[a,b
c,d]

Where Tr(A) = a+d and Det(A) is ad-bc

3. ## Re: Help solving a couple of questions with Linear Algebra

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.