Is there an asymptotic formula for the determinant of a sum of two matrix when one of the matrix is multiplied by some large parameter?

Question

We know that $\operatorname{det}(A + \lambda B) = \operatorname{det}(A)(1 + \lambda \operatorname{Tr}(A^{-1}B) + O(\lambda^2))$ when $\lambda$ is small. What about when $\lambda$ is large? Is there something we can say about the behavior of $\operatorname{det}(A + \lambda B)$? Thanks in advance.

Answer 1

$$\det(A+\lambda B)=\lambda^n\det(\lambda^{-1}A+B)$$ and apply the formula for small $\lambda$.