Let $I$ be an $n\times n$ identity matrix, $B$ be an $n\times n$ matrix with all the elements tending to zero. If we can expression $det(I+B)= 1 + f(B) + o(f^2(B))$, as all the elements of $B$ go to zero, where $f(B)$ is the first order expansion term, what is the exact form of $f(B)$? I have been searching for the answer to this question for very long...I know there are some inequalities but I'd like to know the exact form of $f(B)$, if it exists.



Exact first order perturbation expansion of matrix determinant
The exact form is $f(B) = \mathrm{tr\ } B.$


