TITLE: What is the intuitive meaning of the adjugate matrix?
QUESTION [19 upvotes]: The definition of the adjugate matrix is easy to understand, but I have never seen it used for anything.

What is the intuitive meaning of this matrix?
Are there examples of applications which may shed light on its conceptual meaning?  I would be especially interested to hear examples of usage in representation theory or other abstract algebra disciplines.

REPLY [10 votes]: You have never seen it used for anything? 
The fact that $A\text{adj}(A)=\text{adj}(A)A=\det(A)I$ is the standard one-half of the proof that $A\in\text{GL}_n(R)$ iff $\det(A)\in R^\times$, where $R^\times$ is the group of units of $R$.
The conceptual meaning of the adjugate matrix is somewhat complicated. Really, you can imagine it as being the adjoint of $\bigwedge^{n-1} A$ with respect to a somewhat natural pairing. More information can be found here.