What is the intuitive meaning of the adjugate matrix?
Solution 1:
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.