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?

Cramer's Rule can be used for fitting polynomials to a set of points.
Say we have $n$ ordered pairs $(x_k, y_k)$ with $0<k\le n$ with no $y_k$ the same and we want to find the polynomial $P(x)$ that fits these points. Now we want to find the coefficients of this polynomial but we do know $n$ different values it holds for $n$ different values of $x$.
$$
\left(
\begin{matrix}
x_n^{n-1}&x_n^{n-2}&\cdots&x_n&1\\
x_{n-1}^{n-1}&x_{n-1}^{n-2}&\cdots&x_{n-1}&1\\
\vdots&\vdots&\ddots&\vdots&\vdots\\
x_1^{n-1}&x_1^{n-2}&\cdots&x_1&1
\end{matrix}
\right)
\left(
\begin{matrix}
a_1 \\ a_2 \\ \vdots \\a_n
\end{matrix}
\right)
=
\left(
\begin{matrix}
y_1\\y_2\\ \vdots \\ y_n
\end{matrix}
\right)
$$
Remember that a polynomial of degree $n$ has $n-1$ infliction points. On the first matrix going along we have the same value of $x_k$ at the different powers finally finishing at $0$. When it is multiplied with the column vector the coefficients of $P$ are distributed along the columns, first column being $a_1x_k^{n-1}$ and so on. Now this is a form that can directly be used with Cramer's rule and the coefficients $a_1, ...,a_n$ make up the polynomial $P(x)=\sum^n_{k=1}a_kx^{n-k}$. 
