Any connection between the adjoint map that has determinant $det(\phi)^{(n-1)}$, and the adjoint map that has determinant $ det\phi$?

Question

Is there any connection between the adjoint mapping that is introduced while studying the matrices, and the adjoint mapping that is introduced while studying inner product spaces ?

I mean, for example Greub first define adjoint mapping while in the "Matrices" section as

and after that in the inner product space section,

but clearly these two map are totally different (just check their determinant), but nevertheless they bear the same name, so it there any connection (for a given fixed $\phi$) these to maps ?