I searched through literature but could not find any related topic for the question below. I hope some of you may be able to point me to the right direction.
Let $X: \mathbb{R}\rightarrow\mathbb{R}^{n\times n}$ be a family of $n\times n$ matrices $X(t)$ indexed by $t\in\mathbb{R}$. Assume that we know the determintant of $X(t)$ for all $t$. Is there a way to study the spectrum of matrix $X(0)$ at $t=0$ through the determinant function $\text{det}:t\mapsto\text{det}(X(t))$?
When $X(t) = A - tI$ for some matrix $A$ and identity matrix $I$ then $\text{det}(X(t))$ is the characteristic polynomial of $A$. Therefore knowing $\text{det}(X(t))$ means that we know its zeros or spectrum of $A=X(0)$. Thus, by studying $\text{det}(X(t))$ we can completely characterize the spectrum of $A$. For more general $X(t)$, under what conditions can we compute or infer about eigenvalues of $X(0)$ by using $\text{det}(X(t))$?