When we have a system of of $n$ linear equations represented by $$A \vec{x} = \vec{b} $$ with $\vec{x} = (x\_{1}, x\_{2}, \dots, x\_{n})^{\intercal} $, we can solve for each component of this vector by means of Cramer's Rule:
\begin{equation}
\tag{1}\label{1}
x\_{i} = \frac{\det(A\_{i})}{\det(A)}.
\end{equation}
Here, $A\_{i}$ is formed by replacing the $i$'th column of the column vector of $A$ by the vector $\vec{b}$.


I'm curious whether a similar rule exists for “[continuous](https://math.stackexchange.com/questions/2024034/matrices-with-continuous-indices)” matrix equations. In that regard, we could consider the [Fredholm equation of the first kind](https://en.wikipedia.org/wiki/Fredholm_integral_equation#Equation_of_the_first_kind):
$$g(t) = \int\_{-\infty}^{\infty} K(t,s) f(s) ds ,$$ where $K(\cdot)$ is the
[kernel](https://en.wikipedia.org/wiki/Integral_transform). When $K(t,s) = K(t-s)$, the solution is given by
\begin{equation}
\tag{2}\label{2}
f(s) = \int\_{-\infty}^{\infty} \frac{\mathcal{F}\_{t} [g(t)](\omega) }{\mathcal{F}\_{t} [K(t)](\omega) } e^{2\pi i \omega s} d \omega.
\end{equation}
Here, $\mathcal{F}\_{t}$ is the Fourier transform.


My **questions** are:


1. Can \eqref{2} be considered to be a continuous analogue of \eqref{1}, when one sets $s = s^{\*}$ (a particular value of $s$)?
2. If not, what *is* a continuous analogue of Cramer's rule for continuous matrix equations (involving integration against a kernel)?


N.B. I asked a [similar question on MSE](https://math.stackexchange.com/questions/4094214/is-the-solution-to-the-fredholm-integral-equation-of-the-first-kind-a-continuous).



Is the solution to the Fredholm integral equation of the first kind a continuous analogue of Cramer's rule for matrix equations?
Denote the Fredholm determinant $\Delta=\det(I+K)$, then the solution of the Fredholm equation $g=(I+K)f$ is given formally by
$$f=(I+K)^{-1}g=\frac{1}{\Delta}\left(\frac{\partial\Delta}{\partial K}\right)^\dagger g.$$
This can be seen as the infinite-dimensional, or "continuous", version of Cramer's rule.  

For a derivation, see for example [this review of McKean - *Fredholm determinants*](https://eudml.org/doc/269534).


