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 2: In the formula, $A$.adj$(A)=det(A).I$, the term $adj(A)$ should be understood as adjoint or adjugate?

• In question 2, $adj(A)$ is the adjugate of $A$. – Ofir Schnabel Jul 28 '15 at 9:45

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.