It is well known that given a $n \times n$ matrix $A$, it holds that

$$ \det(I + \varepsilon A)= 1 + \varepsilon \operatorname{tr}(A) + O(\varepsilon ^2).$$

I would need a full representation of $ \det(I + \varepsilon A)$, or at least some bounds explicitly depending on $A$. The reason is that I have to deal with a matrix $A$ which will depend on $\varepsilon$, in particular its entries will have to tend to infinity as $\varepsilon \to 0$ while $\operatorname{tr}(A)$ stays bounded. The problem is that, a priori, higher order terms could mess up the estimate, so I need a more precise representation. A similar question was asked here https://math.stackexchange.com/questions/1174639/series-expansion-of-the-determinant-for-a-matrix-near-the-identity, but without a source. According to the answerer, such formulas are well known by representation theorists (that's the reason of the tag), but he couldn't remember a reference.