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.
1 Answer
$\begingroup$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.
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