Det

$$ \det(AB)=\det(A)\cdot \det(B) $$ $$ \det(A^{-1})=\frac{1}{\det(A)} $$ $$ (\boldsymbol{\sigma}\cdot\boldsymbol{A}) (\boldsymbol{\sigma}\cdot\boldsymbol{B})= \boldsymbol{A \cdot B} + i\boldsymbol{\sigma}\cdot(\boldsymbol{A \times B}) $$ Determinant is the product of all eigenvalues, while Trace is the sum of the eigenvalues.Therefore, $$ \det(\exp(A))=\exp(\mathrm{tr}(A)) $$ Need to be verified.