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Say that a triple of real numbers $(a,b,c)$ is a realizable triple if there are matrices $A,B\in SL_2(\mathbb{R})$ such that $tr (A)=a$, $tr (B)=b$, and $tr (AB)=c$. Question: what is the shape of the ...

Let $A$ and $B$ be two $4\times 4$ matrices. Using Newton's identities, one can prove that if $$\det(A) = \det(B)\quad \text{and}\quad \mathrm{tr}(A^i) = \mathrm{tr}(B^i)$$ for $i=1,2,3$, then $A$ and ...