Let $A\in Mat(\mathbb{R}_{2\times 2})$ such that $A$ is non-inverse matrix and $tr(A)=0$.
Find conditions such that $A_{1,1}=f(A_{2,1},A_{2,2}),A_{1,2}=f(A_{2,1},A_{2,2})$ and find $\frac{dA_{1,2}}{dA_{2,1}}$.
$A$ is non-inverse matrix thus $|A|=A_{1,1}A_{2,2}-A_{2,1}A_{1,2}=0$.
$tr(A)=A_{1,1}+A_{2,2}= 0 $
I consider implicit-function-theorem is useful here but I a don't know how to approach the problem.
Appreiciate any help.