Pensieve header: $\det A=\exp\operatorname{tr}\log A$.

rdet[A_?MatrixQ]:=Module[{n=Length@A},
Normal[Series[Exp[(∑_(k=1)^n)((-Tr[MatrixPower[IdentityMatrix@n-A,k]](ϵ^k))/k)],{ϵ,0,n}]]
]

rdet[(({{a, b}, {c, d}}))]

Simplify[%/.ϵ->1]

Expand[rdet[Table[(Subscript[a, 10i+j]),{i,3},{j,3}]]/.ϵ->1]