Finding determinant from trace and $A^2-4A+3I =0$. Let $A$ and $B$ are two $3 \times 3$ matrices such that $A^2-4A+3I = B^2 - 4B+3I = 0$. Also it is given that $tr(A) = 7$ and $tr(B) = 5$. 
Can we conclude anything about determinant of $A$ and $B$?

Each eigenvalue of $A,\,B$ is $1$ or $3$. So $A$ has eigenvalues $1,\,3,\,3$ of product $9$; similarly, $\det B=\det(1^23)=3$.
