The determinant of a matrix exponential? So I know that if a matrix $A = e^{B}$, then $det(A) = e^{tr(B)}$. I'm wondering, is the converse true? Right now I have a matrix and I know its determinant is $e^{tr(B)}$. So can I conclude that my matrix is $e^{B}$?

No.
For example if $B = \left[\begin{smallmatrix} 1 & 0 \\ 0 & 1 \end{smallmatrix}\right]$ then $e^B = \left[\begin{smallmatrix} e & 0 \\ 0 & e \end{smallmatrix}\right]$ has determinant $e^{\mathrm{tr}(B)} = e^2$.  But $A = \left[\begin{smallmatrix} e^2 & 0 \\ 0 & 1 \end{smallmatrix}\right]$ also has determinant $e^2$ and $A \neq e^B$.
