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.