For $T\in \mathcal L(V)$, we have $\text{adj}(T)T=(\det T)I$. Let $V$ be an $n$-dimensional vector space over a field of characteristic $0$. For a linear operator $T\in \mathcal L(V)$, we know that $\bigwedge^n T=(\det T)I$, where $I:V\to V$ is the identity map.
Further, from this answer we can define the adjugate of $T$ as $\bigwedge^{n-1}T^t:\bigwedge^{n-1}V^*\to \bigwedge^{n-1}V^*$, where $T^t$ is the transpose of $T$. We write $T^\sharp$ as a shorthand for $\bigwedge^{n-1}T^t$.
The Question: It is a well-known formula that if $M$ is an $n\times n$ matrix with entries from a field $F$, then 
$$\text{adj}(M)M=M(\text{adj}(M))=(\det M)I_n$$
where $\text{adj}(M)$ is the adjugate of $M$.
I am trying to formulate this fact in the language of linear maps rather than matrices.
The problem is that it does not mean anything to take the product of $T^\sharp$ with $T$. We just need to make a connection between $T^\sharp$ , $T$, and $\bigwedge^n T^t$.
 A: So let's assume that $V$ has a non-degenerate bilinear form $\langle\cdot,\cdot\rangle$ with a basis $e_1,\dots,e_n$ such that $\langle e_i,e_j\rangle = \delta_{ij}$, the Kronecker delta.  Let $*$ denote the Hodge star operator.  Note that we have the formula
$$ \langle x,y\rangle = *((*x)\wedge y) .$$
Let's identify any operator on $V$ with its matrix representation.  Then we have the following identity:
$$ \langle Tx,y\rangle = \langle x,T^T y\rangle \quad (x,y \in V) .$$
Now we extend $T$ to all of $\Lambda(V)$ in the standard way by the formula $T(x\wedge y) = Tx \wedge Ty$.  Then we have the identities
$$ \text{adj}(T)^T x = *(T(*x)) \quad (x \in V) ,$$
$$ \det(T) = *(T(*1)) ,$$
noting that $*1 = e_1\wedge e_2 \wedge \cdots \wedge e_n$.
Then for all $x,y \in V$, we have
$$ \langle \text{adj}(T)^T x,Ty\rangle = *(T(*x) \wedge Ty) = *(T((*x)\wedge y)) = \det(T) \langle x,y\rangle .$$
Since $\langle\cdot,\cdot\rangle$ is non-degenerate, we have
$$ \det(T) I = (\text{adj}(T)^T)^T \cdot T = \text{adj}(T) \cdot T .$$
A: I'll propose to you another (slightly different, but isomorphic) definition of the adjugate (classical adjoint). Im borrowing from section 8 of http://people.reed.edu/~jerry/332/27exterior.pdf .
Let $f:V\rightarrow V$ (with $n$ the dimension of $V$). We have a canonical isomorphism $\phi:V=\wedge^1 V\rightarrow\mathrm{Hom}(\wedge^{n-1} V,\wedge^n V)$ induced by the Wedge product. Let the adjugate $\mathrm{adj}(f):V\rightarrow V$ of $f$ be obtained from $\mathrm{Hom}(\wedge^{n-1} f,\wedge^{n} V)$ via $\phi$, i.e. $\phi\circ \mathrm{adj}(f)=\mathrm{Hom}(\wedge^{n-1} f,\wedge^{n} V)\circ\phi$. It is then easy to check that $\mathrm{adj}(f)\circ f=\det(f)\mathrm{id}_V$: simply check $\phi((\mathrm{adj}(f)\circ f)v)=\phi(\det(f)v)$ for every $v\in V$.
