The adjugate matrix of an $n \times n$ matrix $A$ is defined by $(\mathrm{adj}\ A)_{k\ell} = (-1)^{k+\ell}\,\det M(\ell,k)$, where $M(\ell,k)$ is the minor matrix obtained from $A$ by deleting row $\ell$ and column $k$.
The obvious application of the adjugate is the identity $A (\mathrm{adj}\ A) = (\mathrm{adj}\ A)A = (\det A)I$. Arguably this gives ample motivation, but only after a 'rabbit-from-hat' definition.
Is there a good way to motivate the adjugate matrix in a first course on matrix algebra?