Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 1 / 24H / ii

The coefficients of $\det (TI-A)$ are polynomial functions of the matrix entries. Its polynomial discriminant $\Delta (A)$ is therefore a polynomial in those entries, and it is nonzero exactly when all roots of the characteristic polynomial are distinct. Thus the required set is the principal open subset $D(\Delta )$ of affine space. It is nonempty: an algebraically closed field is infinite, so choose a diagonal matrix with $n$ pairwise distinct entries.

It is also affine as a variety. The map $A\mapsto (A,1/\Delta (A))$ identifies it with the closed hypersurfaceProjection to $A$ is its regular inverse. This proves the requested affineness while retaining the open embedding into matrix space.