The so-called MacMahon master theorem (also called Wronski identity) and its generalizations (q,super, ...) e.g. Zeilberger et. al. and e.g. Foata et. al. can be
"best" (to my taste) proved via the Koszul duality Koszul algebras and the quantum MacMahon Master Theorem, and also Toru Umeda, Application of Koszul complex to Wronski relations for U(gl).
The main example of Koszul duality is duality between exterior algebra and polynomial algebra, it is enough to obtain the classical MacMahon identity as well as its generalization to certain matrices with non-commuting entries (Manin matrices). There are further q-super analogs - everything is the same just need to consider appropriate versions of the exterior algebra.
Let me sketch the idea of application which is quite simple, it is related to Vladimir's answer above.
One knows that Euler characteristic can be calculated from complex itself and from (co)homology of the complex - we get the same result. This actually can be generalized to the trace of arbitrary operator acting on the complex, if it commutes with differential.
So the proof of the MacMahon formula exploits this idea for the Koszul complex, which cohomology consits of $C$. So the trace taken over cohomology is $1$, while the trace calculated via the complex itself will give $det(1-A)(\sum_k Tr S^k A)$.
The advantage of this proof that it can be generelized to matrices with non-commuting entries - Manin matrices, quantum matrices, super matrices, etc... While the standard proof while via the diagonalization will not work in such situations.
This also has some applications in representation theory and quantum integrable systems The MacMahon Master Theorem for right quantum superalgebras and higher Sugawara operators for \hat gl(m|n) A. I. Molev, E. Ragoucy;