Semi-Invariants of a Matrix and Covector

We prove the following theorem: let ${\mathcal{M}}_d$ denote the set of $d\times d$ matrices over an infinite field $K$, and let $(K^d{)}^{\ast }$ be the set of row vectors. Define an action of ${\mathrm{S}\mathrm{L}}_d(K)$ on $X:={\mathcal{M}}_d\oplus (K^d{)}^{\ast }$ by $$g\cdot (A,\phi )=(gA{g}^{-1},\phi g^{-1})\mathrm{.}$$ Then $K[X{]}^{{\mathrm{S}\mathrm{L}}_d}$ is a polynomial ring, generated by the coefficients of the characteristic polynomial of $A$ and one further invariant, namely $\mathrm{\Delta }(A,\phi ):=\det (\phi ,\phi A,\phi A^2,\dots ,\phi A^{d-1}{)}^t\mathrm{.}$ Our proof is entirely classical in nature, but we give an interpretation of the result and its proof in terms of quiver representation theory.