The notions of adjoint and adjugate, which I saw, are as follows:
(1) Let $T:V\rightarrow W$ be a linear map. Then there is a corresponding linear map between the duals of these spaces: $T^*:W^*\rightarrow V^*$, defined as follows: for every linear map $f$ from $W$ to $k$ (the field), there is a linear $f^*$ from $V$ to $k$ given by $f\circ T$, hence we have a map $f\mapsto f^*$ from $W^*$ to $V^*$.
The map $T^*$ is called the adjoint of $T$.
(2) The notion of adjugate of a matrix is defined as follows: given a square matrix $A$, the transpose of the matrix of cofactors of $A$ is called the adjugate of $A$.
Question 1: Are these notions of adjoint and adjugate related?
Question 2: In the formula, $A$.adj$(A)=det(A).I$, the term $adj(A)$ should be understood as adjoint or adjugate?
Best Answer
Not really.
The adjoint operator can be defined for arbitrary topological vector spaces; adjugate requires finite-dimensioned spaces.
The adjoint operator is, for real matrices, just the transposed matrix. The adjugate, as you see, has an entirely different definition.
As per your question $2$, $adj(T)$ is the adjugate operator.