# The difference between adjoint in linear algebra and adjoint of operator?

I'm really confused about the terminology of adjoint.

Where in operator $A$ and $B$ was said to be adjoint iff $\langle Ax,y\rangle =\langle x,By\rangle$; yet in linear algebra adjoint of a matrix, which was also called adjugate of the matrix, was defined by cofactors.

Thus, in operator $adj(adj(A))=A$ because of the involutiveness.

Yet in linear algebra, (adjugate) $adj(adj(A^\prime))=det|A^\prime|^{n-2}A^\prime$ by computation.

Clearly, there seemed to be some connection between those two terminologies.

My questions were:

1. What's the definitive connection?

2. How am I suppose to avoid mistakes, and distinguish the usage of those two terminologies?

*Can assume $A$ is a bounded linear operator from a Hilbert space, definition as given.

Notes: I saw a previous example Adjoint and Adjugate are same or different? where it was answered in a way such that the definition was different. However, since operator could be presented in its matrix form and matrix could be seen as an operator, I'd like to know:

1. Under which minium circumstance those two terminologies were equivalent.

2. If there is any way to present one of the terminologies in terms of the combination of another. For example, a silly case in matrix form when $A$ was constant $Adjugate(Adjugate(A))=det |A|^{n-2}A=det |A|^{n-2}adj(adj(A))$