# What is the intuitive meaning of the adjugate matrix?

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.

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The fact that $A\text{adj}(A)=\text{adj}(A)A=\det(A)I$ is the standard one-half of the proof $A\in\text{GL}_n(R)\Leftrightarrow \det(A)\in R^\times$.
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 (page 432).
@Alex_Youcis What does the symbol $R^\times$ mean? –  user1551 Dec 9 '12 at 19:23
The group of units of $R$. –  JSchlather Dec 9 '12 at 19:23