What is Cramer's rule used for? Cramer's rule appears in introductory linear algebra courses without comments on its utility.  It is a flaw in our system of pedagogy that one learns answers to questions of this kind in courses only if one takes a course on something in which the topic is used.
On the discussion page to Wikipedia's article on Cramer's rule, we find this detailed indictment on charges of uselessness, posted in December 2009.
But in the present day, we find in the article itself the assertion that it is useful for


*

*solving problems in differential geometry;

*proving a theorem in integer programming;

*deriving the general solution to an inhomogeneous linear differential equation by the method of variation of parameters;

*(a surprise) solving small systems of linear equations.  This one is what it superficially purports to be in linear algebra texts, but then elementary row operations turn out to be what is actually used.


At some point in its history, the Wikipedia article asserted that it's used in proving the Cayley–Hamilton theorem, but that's not there now.  To me the Cayley–Hamilton theorem has always been a very memorable statement, but at this moment I can't recall anything about the proof.
What enlightening expansions on these partial answers to this question can the present company offer?

If you are asking for other applications of Cramer's rule, here is an application I learnt about recently in Harold's book Galois theory:
In Harold's book, Cramer's rule is used to prove Lagrange's theroem (not the one about groups).
Let $K$ be a field, $n$ be a positive integer. The following theorem will be formulated using the obvious action of $S_n$  on $K[x_1,\ldots,x_n]$.
Lagrange's Theorem:
Let $P,Q\in K[x_1,\ldots,x_n]$, then: $\operatorname{Stab}(P)\subseteq \operatorname{Stab}(Q)$ (Stab is the stabilizer subgroup) iff there exists symmetric polynomials $s_0,s_1,\ldots,s_a,t_0,t_1,\ldots,t_b\in K[x_1,x_2,\ldots,x_n]$ such that:
$$Q=\frac{s_aP^a+s_{a-1}P^{a-1}+\cdots+s_0}{t_bP^b+t_{b-1}P^{b-1}+\cdots+t_0}$$
